Random and Vector Measures:
From \Toy" Measurable Systems to Quantum Probability
by
Kristin Courtney
A thesis submitted to the Graduate Faculty of
Auburn University
in partial ful llment of the
requirements for the Degree of
Master of Science
Auburn, Alabama
August, 3 2013
Approved by
Jerzy Szulga, Professor of Mathematics and Statistics
Olav Kallenberg, Professor of Mathematics and Statistics
Bertram Zinner, Associate Professor of Mathematics and Statistics
Georg Hetzer, Professor of Mathematics and Statistics
Abstract
Vector measure theory and Bochner integration have been well-studied over the past
century. This work is an introduction to both theories and explores various examples and
applications in each. The theories and theorems are pre-existing, whereas the examples and
discussions are mine. Our primary examples of vector measures are toy vector measures,
which serve as a class of elementary yet nontrivial structures that enables us to grasp the
spirit and essence of the advanced theory, both on the conceptual and technical level. We
also discuss random measures as special cases of vector measures.
The theory of Bochner integration is introduced as a framework for the Radon-Nikod ym
Property, which comes from the failure of the Radon-Nikod ym Theorem to hold when gen-
eralized to Banach spaces. The consequences of this failure as well as Rie el?s extension of
the theorem are discussed in Chapter 2.
Finally, we conclude with a brief introduction to Hilbert quantum theory and quantum
probability and introduce possible vector extensions of quantum probability theory.
ii
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
1 Vector Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 History and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Vector Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 \Toy" Vector Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Variation and Semivariation . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Linear Operators, Integrals with respect to Vector Measures, and descriptive
theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6 Vector Measures and Random Measures . . . . . . . . . . . . . . . . . . . . 15
1.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 Bochner Integration and the RNT . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1 Bochner Integral for -Simple Functions . . . . . . . . . . . . . . . . . . . . 21
2.2 Bochner integral de ned . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 Bochner?s Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Pettis Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5 Extending R results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.6 The Radon Nikod ym Property . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.6.1 Signi cant Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.7 Radon Nikod ym Theorem for Bochner Integration . . . . . . . . . . . . . . . 43
3 Quantum Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1 Hilbert Space Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Basics of Quantum Probability . . . . . . . . . . . . . . . . . . . . . . . . . 52
iii
3.2.1 Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.2.2 Intersections and Unions . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2.3 A Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2.4 Observables as Random Variables . . . . . . . . . . . . . . . . . . . . 58
3.3 Quantum Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.4 Quantum Vector Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.5 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
A Measure Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
A.1 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
A.2 Integration and Noteworthy Theorems . . . . . . . . . . . . . . . . . . . . . 64
B Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
B.1 Measure Theory and Probability Theory . . . . . . . . . . . . . . . . . . . . 66
B.2 Other Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
C Radon-Nikod ym Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
C.1 Radon Nikod ym Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
C.2 Other Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
C.3 Some Signi cant Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 69
C.3.1 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
C.3.2 Conditional Expectation . . . . . . . . . . . . . . . . . . . . . . . . . 70
D Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
D.1 Some Signi cant Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
D.1.1 Example Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
D.2 Lp Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
D.2.1 De nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
D.2.2 Lp for 0 p< 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
D.3 Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
iv
D.3.1 Dual Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
D.3.2 More on Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
D.4 Operators in a Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
E Summability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
F Harris Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
G Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
v
List of Figures
2.1 T : [0;1]!R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2 b : R!R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3 b1 : [0;1]!R and b2 : [0;1]!R . . . . . . . . . . . . . . . . . . . . . . . . . . 30
D.1 Unit Circle for l22(R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
D.2 Unit Circle for l12(R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
D.3 Unit Circle for l12 (R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
D.4 Unit Circle for lp2(R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
vi
Chapter 1
Vector Measures
1.1 History and Motivation
Measure theory was developed primarily during the late 19th and early 20th century.
By the 1930?s investigations into extending real variable theory to functions taking values
in Banach spaces had begun. Among the noteworthy results were Gel?fand?s use of vector
measure-theoretic techniques to prove that L1[0;1] is not the pre-dual of any Banach space
(1938) and Pettis?s contribution to the Orlicz-Pettis Theorem (weakly countably additive
vector measures are norm countably additive)[2].
Naturally, representing linear operators as integrals of Banach-valued functions (i.e.
Bochner integrals), which, requires some form of a Radon-Nikod ym Theorem, has been of
great interest. Hence, elements of vector measure theory, speci cally elements concerning
what is know as the Radon-Nikod ym Property, are remarkably proli c in theories regarding
classi cation of topological vector spaces, operators, and Banach spaces.
In 1933, Bochner introduced an integral of a Banach-valued function with respect to a
scalar measure in Integration von Funktionen deren Werte die Elemente eines Vectorraumes
sind, which is now called the Bochner integral. In 1936, J.A. Clarkson as well as N. Dun-
ford and A.P. Morse established that absolutely continuous functions on a Euclidean space
with values in a uniformly convex Banach space, or resp., a Banach space with a bound-
edly complete basis, are the integrals of their derivatives.[2] These were later recognized by
Dunford as Radon-Nikod ym theorems for the Bochner integral and hence, the rst of the
Radon-Nkiod ym theorems for vector measures. [2] In 1943, Phillips extended the Dunford
and Pettis? Radon-Nikod ym Theorem.[11] In 1968 Rie el published the strongest version of
the Radon-Nikod ym Theorem for the Bochner integral.
1
In the 1960?s several results emerged in connection to Martingales, which lead to various
purely geometrical characterizations of spaces with the Radon-Nikod ym Property; one of
the best known results is from Chatterji (1968)[1] pertaining to the relationship between the
Radon-Nikod ym Theorem and the martingale mean convergence theorem. In the 1970s, work
in di erentiating vector measures lead to geometric results involving the Radon-Nikod ym
Property in Banach spaces.[18] In 1983, R.D. Bourgin showed that the Radon-Nikod ym
Property is equivalent to several convergence properties for Bochner-valued martingales.[18]
In the rst chapter, we will give a brief introduction to vector measures as well as
an overview of several major results in vector measure theory. In the second chapter, we
will progress to Bochner integration and the Radon-Nikod ym Property with some attention
paid to Pettis integration and the general Radon-Nikod ym theorem for Bochner integrals.
In the nal chapter, we will introduce an axiomatic approach to quantum mechanics using
so-called \Quantum Probability" and formulate a hypothesis on the existence of \quantum
vector probabilites".
1.2 Vector Measures
Let F0 be a ring of subsets of a set , F a -ring of subsets of , 0 an algebra on ,
and a -algebra on ,1 Also, let X be a vector space over the eld K(= C or R).
De nition 1.1. A vector-valued function F :F0 !X is called additive if, given disjoint
A;B2F0, then F(A[B) = F(A) +F(B).2
This is well-de ned since X is a vector space. If we assume that X is a topological vector
space, then we can consider countable additivity as well.
De nition 1.2. Let X be a topological vector space F :F!X is countably additive if,
for all sequences (Ek) of pairwise disjoint members of F for which
1[
k=1
Ek2F,
1These terms are de ned in Appendix A.
2Since ;\; = ;, this condition also implies that (;) = (;[;) = (;) + (;) = 2 (;), and hence
(;) = 0.
2
F
1[
k=1
Ek
!
=
1X
k=1
F(Ek),
in the topology on X.
The rst two immediate examples of vector measures are signed measures and complex
measures from basic measure theory.
Example 1.3. Consider the vector space R and measurable space ( ; ). Then a vector
measure F : !R is called a signed measure on ( ; ).
Example 1.4. Similarly, if the vector space is C, then a vector measure F : !C is called
a complex measure on ( ; ).
The equation for countable additivity must hold in the topology of X in order for F to
be well-de ned. This means that, not only must the series converge, but, since the set union
1[
k=1
Ek is invariant under permutations of the Ek?s, the series must converge unconditionally.
To understand the strength of this requirement, consider a sequence (Ek) of pairwise disjoint
members of F for which
1[
k=1
Ek 2F. If F is well-de ned, then F
1[
k=1
Ek
!
= x0 for some
x0 2 X. Since F is nitely additive, for all n, F
n[
k=1
Ek
!
=
nX
k=1
F(Ek). For each n,
de ne xn 2 X as xn :=
nX
k=1
F(Ek). Then, if F is well-de ned, xn ! x0 in X. Hence,
F
1[
k=1
Ek
!
= limn!1
nX
k=1
F(Ek) =
1X
k=1
F(Ek). Since the series converges conditionally, i.e.
with any permutation of the F(Ek)?s, the series will also converge to x0 in X with respect
to the topology on X.
Therefore, a general topological space (or even a metric space) is not enough. Moreover,
theorems about such spaces would be too few to be of interest of in depth study. Hence,
for any reasonable purposes, all of the image spaces are F-spaces, which ensure luxury of
addition, topological convergence, and a triangle inequality.
3
De nition 1.5. An additive vector-valued function from a -ring F into a topological
vector space X is called a vector measure on ( ;F), and a countably additive vector-
valued function from a -ring F into a topological vector space X is called a countably
additive vector measure on ( ;F).
Example 1.6. Let p2[0;1), and de ne F : ([0;1];B; )!Lp([0;1];B; ) by F(E) := 1IE
for all E 2B. Then, F is nitely additive and also countably additive by the countable
additivity over domain of integration for the Lebesgue integrals 3.
Example 1.7. Consider the previous example but with p = 1. F is still nitely addi-
tive; however, F is not countably additive: let fEngn2N be a pairwise disjoint collection
of measurable sets with positive measure, and let E := SnEn. Then, F(E) = 1IE, and
P
nF(En) =
P
n 1IEn. But for all n2N,
k1IE
nX
k=1
1IEkk1 =k1IE 1ISnk=1Ekk1 =k1IS1k=n+1Ekk1 = 1.
Therefore, F(E)6= PnF(En).
Note that some texts use \vector-measure" to describe a countably additive vector-
valued set function. Other texts reserve the name \vector-measure" for countably additive
set functions that take values in a Banach space.4 This is due, in large part, to the fact that
Banach spaces admit a plethora of linear functionals and linear operators, which are crucial
tools in characterizing vector measures. We will also make frequent use of the dual of our
image space, X , which will be desirably rich if X is a Banach space (or at least locally
convex).
1.3 \Toy" Vector Measures
The term toy vector measures comes from Paul-Andre Meyer?s term toy Fock spaces in
his book Quantum Probability for Probabilists,[12] in which the term was used to encompass a
3and because, for all p2[1;1) and all E2B, j1IEjp = 1IE.
4However, many of the theorems we will see in this chapter to not require that our vector measures be
countably additive.
4
class of elementary yet nontrivial structures to grasp the spirit and essence of the advanced
theory, both on the conceptual and technical level. We have adopted this phrasing for a
similar purpose.
Let X be an F-space5 and (N;2N;c) a measure space. Fix a sequence (xn)1n=1 2XN,
and de ne F : 2N!X by F(E) :=
X
n2E
xn for all E N.
Before we discuss whether or not F is a vector measure, we must rst determine whether
or not it is well-de ned. (In fact, in doing so, we will have established both nite and
countable additivity.) If E N is nite, then
X
n2E
xn 2X. Suppose, then, that E is not
nite. For the sake of simplicity, we will consider the case E = N.
As mentioned before, since N = Snfng, the well-de ned claim reduces to the claim that
the series
X
n2N
xn converges unconditionally. Since X is an F-space, Orlicz Theorem gives us
useful characterizations of unconditional convergence:
Theorem 1.8. In a complete metrizable topological vector space, the following are equiva-
lent:6
1. (xk) is summable, that is, for every > 0, there is a nite set K N such that, for
every nite L N that is disjoint with K,
X
n2L
xk
< .
2.
X
k2N
xk converges unconditionally in X.
3.
X
k2N
skxk converges for every sequence (sk)2f 1;1gN.
.
X
k2N
skxk converges for every sequence (sk)2f0;1gN.
4. If X is also a Banach space, then (xk) 2XN is summable i
X
k2N
skxk converges for
any bounded sequence (sk) of scalars.
5i.e. a complete metrizable topological vector space with a translation-invariant F-norm kxk= d(x;0).
6The origin of the particular formulations is hard to nd as these claims have been well-integrated into
the subject of functional analysis.
5
Furthermore, in even an F-space, absolute convergence with the triangle inequality
immediately imply summability. However, only in nite dimensional Banach spaces are
the two equivalent[5]. (The reverse implication comes from the fact that, in such spaces,
convergence is equivalent to coordinatewise convergence.) An easy counterexample in the
nite dimensional case is the series
1X
n=1
1
nen where (en) is the standard basis of the vector
space of all sequences with nitely many non-zero terms. The proofs of the preceeding
theorems may be found in the appendices.
Therefore, F is a countably additive vector measure if and only if the corresponding
sequence (xn) is summable.
1.4 Variation and Semivariation
An important concept in vector measures is variation, which acts similarly to a norm
for our vector measures7
De nition 1.9. Let F : !X be a vector measure and X an F-space. The variation of
F is the extended nonnegative function jFj: ! [0;1] given for all E2 by
jFj(E) = supf
nX
k=1
jjF(Ek)jjX :
n[
k=1
Ek E and Ek2 are pairwise disjoint 1 k ng.
If jFj( ) <1, then F is called a measure of bounded variation. If we require that
F be - nite on ( ; ), jFj : ! [0;1] on ( ; ), will be contably additive, and hence a
non-negative R-valued measure.
Example 1.10. Supposing that X is a Banach space, consider bounded variation for our
toy vector measure, F : 2N !X, given by F(E) :=
X
n2E
xn for all E N. (where (xn) is
summable). Let E N and fEkgmk=1 a pairwise disjoint collection subsets of N such that
m[
k=1
Ek E. Then,
mX
k=1
jjF(Ek)jj=
mX
k=1
X
j2Ek
xj
mX
k=1
X
j2Ek
kxjk=
X
j2Smk=1Ek
kxjk
X
j2E
kxjk.
7In fact, we can de ne a norm on vector measures on ( ; ) by kFk:=jFj( ).
6
Hence, jFj(E)
X
j2E
kxjk for all E2 ; that is, F is guaranteed to have bounded variation
if (xj) is absolutely convergent.8
Conversely, if F is of bounded variation, then (xn) is absolutely convergent. To see this,
let (xn) be a summable sequence in X so that the F de ned by (xn) is of bounded variation.
Then,
1>jFj(N) sup
m2N
mX
n=1
kF(fng)k
!
+kF(fn2Njn>mg)k
!
sup
m2N
mX
n=1
kxnk
!
=
1X
n=1
kxnk:
Example 1.11 (Vector Measure without Bounded Variation). [2] Consider again the ex-
ample where F : ([0;1];B; ) !L1([0;1];B; ) is given by F(E) := 1IE for all E2B. Let
fEngn2N be a pairwise disjoint collection of measurable sets with positive measure, and let
E := SnEn. For each n2N, let n =fE1;:::;En 1;S1k=nEkgbe a partition of E into nite
disjoint measurable sets each of nite measure. Then for each n2N,
X
A2 n
kF(A)k=
n 1X
k=1
k1IEkk1 +k1IS1k=nEkk1 = n.
Hence, jFj(E) =1 and F is not of bounded variation.
There are numerous interesting vector measures without bounded variation, but we will
save those for the coming section on random measures.
Lemma 1.12. : jFj: !X is a monotone function on .
Proof. Let E;A2 and A E. Then for any pairwise disjoint collection, fAkgnk=1
such that Ak A for 1 k n, Ak E for 1 k n. Then,(
nX
k=1
jjF(Ak)jjX :
n[
k=1
Ak A and Ak2 are pairwise disjoint for 1 k n
)
( nX
k=1
jjF(Ak)jjX :
n[
k=1
Ak E and Ak2 are pairwise disjoint for 1 k n
)
,
and hence jFj(E) jFj(A).
8In fact, kFk= P
nkxkn.
7
Lemma 1.13. jFj: !X is nitely additive.
Proof. It will su ce to show that for E;A2 with E\A =;thatjFj(E[A) =jFj(E) +
jFj(A). Let fBigli=1 be a disjoint collection in such that Bi E[A for 1 i l. Let
Ei := Bi\E and Ai := Bi\A for 1 i l. Then fEigli=1[fAigli=1 is a pairwise disjoint
collection in for which Ei E and Ai A for 1 i l. Then,
lX
i=1
jjF(Bi)jj=
lX
i=1
jjF(Ei[Ai)jj=
lX
i=1
jjF(Ei) +F(Ai)jj
lX
i=1
jjF(Ei)jj+jjF(Ai)jj=
lX
i=1
jjF(Ei)jj+
lX
i=1
jjF(Ai)jj
jFj(E) +jFj(A):
Now, let fEkgnk=1 be a disjoint collection in such that Ek E for 1 k n, and let
fAjgmj=1 be a disjoint collection in such that Aj A for 1 j m. Let Bi = Ei for
1 i n and Bn+i = Ai for 1 i m. Then, since A\E = ;, fBign+mi=1 is a disjoint
collection in such that Bi E[A for 1 i n+m. Then,
jFj(E[A)
n+mX
i=1
jjF(Bi)jj=
nX
k=1
jjF(Ek)jj+
mX
j=1
jjF(Aj)jj.
Taking the supermum over fEkg and fAjg, jFj(E[A) jFj(E) +jFj(A).
Theorem 1.14. If F : !X is a a countably additive vector measure of bounded variation,
then jFj is countably additive.
In other words, jFj is a \true" or classic measure.
Proof. LetfEkgk2N be a countable collection of disjoint measurable sets. If eitherjFj(SkEk) =
1 or PkjFj(Ek) = 1, then the proof is immediate. Suppose, then, that F is of bounded
variation.
1. Countable subadditivity:
8
Let fAjgnj=1 be a disjoint collection of measurable subsets of SkEk. Then for each
k2N, fAj\Ekgnj=1 is a disjoint collection of measurable subsets of Ek and so
jFj(Ek)
nX
j=1
jjF(Aj\Ek)jj. Then,
X
k
jFj(Ek)
X
k
nX
j=1
jjF(Aj\Ek)jj
=
nX
j=1
X
k
jjF(Aj\Ek)jj
nX
j=1
X
k
F(Aj\Ek)
=
nX
j=1
F
[
k
(Aj\Ek)
!
=
nX
j=1
jjF(Aj)jj:
Hence,
X
k
jFj(Ek) jFj
[
k
Ek
!
.
2. Countable superadditivity:
For each n2N,
n[
k=1
Ek
[
k
Ek. Then, by monotonicity and nite additivity of jFj,
for all n2N,
jFj
[
k
Ek
!
jFj
n[
k=1
Ek
!
=
nX
k=1
jFj(Ek).
Hence, jFj
[
k
Ek
!
1X
k=1
jFj(Ek).
Since,jFj(;) = 0, we have thatjFjis a scalar measure on K (and a non-negative scalar
measure in case K = R). If F is of bounded variation, then for any collection fEkgk2N2 ,
1>jFj(
[
k
Ek) = kjFj(Ek). However, if F is not of bounded variation, then we will have
divergent series
X
k
jFj(Ek).
Another important concept for vector measures is weak-variation.
De nition 1.15. Let F : !X be a vector measure and X a Banach space. The weak-
variation of F is the extended nonnegative function jFj : ! [0;1] given by
9
jFj (E) = supfjx Fj(E) : x 2X ;jjx jj 1g,
where jx Fj is the variation of the K-valued measure x F 9, for all E2 .
If jFj ( ) <1, then F is said to be of bounded weak-variation.
Note that for any x 2 X with jjx jj 1, E 2 , and pairwise disjoint collection
fEkgnk=1 such that
n[
k=1
Ek E,
nX
k=1
jx F(Ek)j
nX
k=1
jjx jjjjF(Ek)jj
nX
k=1
jjF(Ek)jj.
Therefore, jFj (E) jFj(E) for all E2 .
For an instance of strict inequality, consider the Brownian Motion X =fXt : t 0g, and
F be given by F(A) = RAf dX, for a bounded measurable f : R+ !R. Then jFj(A) =1
for all non-degenerate intervals A, but jFj (A) <1.[15]
Proposition 1.16. [2] If F : ! X is a vector measure of bounded variation, then a
non-negative R-valued measure : ![0;1] is the variation jFj of F i
1. jx Fj(E) (E) for all E2 and all x 2X with jjx jj 1, and
2. If : ! [0;1] is a measure satisfying jx Fj(E) (E) for all E 2 and all
x 2X with jjx jj 1, then (E) (E) for all E2 .
In other words,jFjis the least upper bound (if it exists) offjx Fj: x 2X andjjx jj 1g.
Proof. ) First, we will show that the two properties hold for =jFj.
Let x 2X and jjx jj 1.
1. jx Fj(E) jFj(E) comes from jFj (E) jFj(E) for all E2 .
2. Let be a measure satisfying the assumptions of the implications in 2. LetfEigni=1 be
a disjoint collection of measurable subsets of E. Then,
9jx Fj(E) = supfPn
k=1jx
F(Ek)j:fEkgn
k=1 ;
Sn
k=1Ek Egis well-de ned since x
F is well-de ned.
10
n[
i=1
Ei
!
=
nX
i=1
(Ei)
nX
i=1
jx Fj(Ei)
for all x 2X with jjx jj 1. Since jjF(E)jj = supfjx Fj(E) : x 2X ;jjx jj 1g
for all E2 , we have that
nX
i=1
(Ei)
nX
i=1
jjF(Ei)jj. Therefore, (E) jFj(E).
( Suppose is a measure on that satis es the two conditions.
Then, by the second condition (with = jx Fj for some xed x 2 X ), we know
(E) jFj(E) for all E2 , and by the fact that jFj (E) jFj(E) for all E2 , we have
the reverse inequality.
Proposition 1.17. [2] Let F : !X be a vector measure, then for all E2 ,
1. jFj (E) = sup
(
nX
k=1
akF(Ek)
: ak2K;jakj 1for1 k n;andfEkg
n
k=1 is a nite partition ofE
)
.
2. sup
E A2
kF(A)k jFj (E) 4 sup
E A2
kF(A)k.
Proof. 1. Let E 2 and fEkgnk=1 a partition of T into pairwise disjoint sets in and
ak2BK10 for 1 k n. Then,
nX
k=1
akF(Ek)
= supx 2BX
x
nX
k=1
akF(Ek)
supx 2BX
nX
k=1
jx F(Ek)j jFj (E):
For the reverse inequality, let x 2X withkx k 1, E2 , andfEkgnk=1 a partition
of T into pairwise disjoint sets in . Then,
nX
k=1
jx F(Ek)j=
nX
k=1
sgn(x F(Ek))x F(Ek)
=
x
nX
k=1
sgn(x F(Ek))F(Ek)
!
nX
k=1
sgn(x F(Ek))F(Ek))
:
2. For the rst inequality, let E2 . Then,
10BX denotes the closed unit ball in the normed vector space X.
11
sup
E A2
kF(A)k= sup
x 2BX
sup
E A2
jx F(A)j kF(E)k.
For the second inequality, we rst assume X is a Banach space over R. Let E 2 ,
fEkgnk=1 a partition of T into pairwise disjoint sets in , and x 2 BX . De ne
N+ :=fk : 1 k n;x F(Ek) 0g and N :=fk : 1 k n;x F(Ek) < 0g. Then,
nX
k=1
jx F(Ek)j=
X
k2N+
x F(Ek)
X
k2N
x F(Ek)
x
X
k2N+
x F(Ek)
!
+
x
X
k2N
F(Ek)
!
2 sup
E A2
kF(A)k
If the scalar eld is C, the following argument will work for x F split into real and
imaginary parts, yielding the 4 sup
E A2
kF(A)k in the proposition.
Note that the second claim implies that a vector measure is of bounded weak-variation i
its range is bounded in X; hence, a vector measure of bounded weak-variation is called a
bounded vector measure.
1.5 Linear Operators, Integrals with respect to Vector Measures, and descrip-
tive theorems
As mentioned before, elements of vector measure theory can be very useful in describing
the Banach spaces into which vector measures map. The following is a slight digression to
illustrate such descriptions.
With weak-variation in hand, we are ready to construct a rudimentary integral of a
bounded measurable function with respect to a bounded vector measure. We follow Diestel
and Uhl?s construction from Chapter 1 of Vector Measures:
12
Let be an -algebra on and F : !Xa bounded vector measure. Let B(( ; );K)
denote the space of all bounded -measurableK-valued function on (with supremum norm
k k1), and let B0(( ; );K) = B0 denote the subspace of all simple scalar functions on .
Then, de ne TF : B0 !X by
TFf :=
nX
k=1
akF(Ek)
for each f2B0 (where f is given by f :=
nX
k=1
ak1IEk for some n2N with ak2K and Ek2
for 1 k n and fEkgnk=1 a partition of E). Then TF is a linear map and, for all f2B0,
jjTF(f)jj=
nX
k=1
akF(Ek)
=jjfjj1
nX
k=1
ak
jjfjj1F(Ek)
jFj
( )jjfjj1.
Then TF has a unique continuous linear extension to the space of all K-valued functions on
that are uniform limits of simple functions -measurable K-valued functions on , B( 0).
Now, we are ready to de ne an integral with respect to our vector measure.
De nition 1.18. Let ( ; ) be a measurable space and F : !X be a bounded vector
measure, the for each f2B( ), we de ne R f dF by
Z
f dF = TF(f).
This is only a crude introduction to integration with respect to vector measures, but,
as it is not in the scope of this paper, a crude introduction will have to do. This integral is,
in fact, linear, and, as indicated above, satis es
Z
f dF
jjfjj1jFj ( ).
Our nal remark on this integral is the following theorem:[2]
Theorem 1.19. Let ( ; ) be a measurable space and X a Banach space, and let : !
[0;1] be a non-negative R-valued measure on ( ; ). Then there is a one-to-one linear
correspondence between L(L1( );X)) and the space of all bounded vector measures F :
!X that vanish on -null sets give by (TF(f)) =
Z
f dF for all f2L1( ).
13
We will close this section with a few more de nitions and important theorems, the
discussions of which we will omit.
De nition 1.20. Let 0 be an algebra on , F : 0 !X a vector measure, and :F!R
a nite measure on F. Then F is called -continuous, F << if lim
(E)!0
F(E) = 0.
De nition 1.21. Let 0 be an algebra on , and F : 0 ! X a vector measure. F
is strongly additive whenever, given a pairwise disjoint sequence (Ek) in 0, the series
1X
k=1
F(Ek) converges in the norm.
If is a -algebra, this is weaker than countable additivity; note that, without an
equation, convergence need not be unconditional.
De nition 1.22. A family fF : 0 !X : 2Tg of strongly additive vector measures
is uniformly strongly additive whenever, for any pairwise disjoint sequence (Ek) 0,
lim
k!1
1X
m=k
F (Em)
= 0 uniformly for 2T.
Again, iffF : 0 !X : 2Tgis a family of countably additive vector measures, then
uniform strong additivity is just uniform countable additivity.
As previously mentioned, many of our theorems will hold for nitely additive vector
measures; however, several require the slighly stronger requirement of strong additivity, such
as the two following theorems.
Theorem 1.23 (Nikod ym Boundedness Theorem). [2] Let ( ; ) be a measurable space and
fF : 2Tg a family of X-valued vector measures de ned on . If sup
2T
jjF (E)jj<1 for
each E2 , then fF : 2Tg is uniformly bounded, i.e. sup
2T
jF j ( ) <1.
Theorem 1.24 (Vitali-Hahn-Saks-Nikod ym Theorem). [2] Let be a - eld of subsets of
and (Fn) a sequence of strongly additive X-valued measures on . If limnFn(E) exists in
X-norm for each E2 , then the sequence (Fn) is uniformly strongly additive.
14
Theorem 1.25 (Orlicz-Pettis). 11[2] Let Pnxn be a series in X such that every subseries
of Pnxn is weakly convergent. Then Pnxn is unconditionally convergent in norm. Con-
sequently, a weakly12 countably addtitive vector measure on a -algebra is (norm) countably
additive.
Finally, we close this section with one of the most powerful theorems for vector (and
scalar)-valued set functions.
Theorem 1.26. [3] Let be a -algebra of subsets of a set , and let be a complex or
signed measure on . Suppose fFng is a sequence of -continuous vector or scalar valued
additive set functions on such that limnFn(E) exists for each E2 . Then,
lim
j j(E)!0
Fn(E) = 0
uniformly for n = 1;2;:::.13
1.6 Vector Measures and Random Measures
One of the most natural examples14 of a vector measure is a random measure. In this
section we will introduce random measures and discuss where they t in vector measure
theory. First, note that there are at least two senses of the term \random measure". The
rst is a wider sense, which we will just mention:
In a wide sense, a random measure is simply a vector measure whose range is L0(S;S;P),
where (S;S;P) is a probability space.15 Since members of L0(S;S;P) are called random
variables, the vector measure is called a random measure. A primary example of a random
measure in the wide sense is \White Noise":
11This is often presented with the Orlicz Theorem given with the Toy Vector Measures example.
12that is, the series converges weakly in X
13It certainly bears mentioning that this theorem is very much akin to a corollary of the Uniform Bounded-
ness Principle, and, in fact, the proofs of the Vitali-Hahn-Saks Theorem (whether they use the gliding-hump
method or Bare Category Theorem) greatly resemble those for the Uniform Boundedness Principle
14\Example" should be used loosely here, for, as we will see, a random measure may not be additive.
15Note that, although L0(S;S;P) is not a Banach space, we can de ne a complete metric on L0(S;S;P),
e.g. the metric given by d(X;Y) = Emaxf1;jX Yjg, and hence we consider L0(S;S;P) an F-space.
15
Example 1.27. Say we have a Brownian motion Xt on ( ;F;P), and let B[0;1] = B
represent the -algebra of sets on [0;1]. ThenB is generated by the eld,B0 of nite unions
of intervals of the form (a;b] (0;1].
De ne X : f(a;b] : (a;b] (0;1]g ! L2( ;F;P) by X(a;b] = Xb Xa, i.e. the
increments of Xt.
For A 2B0 where A is the nite union of pairwise disjoint intervals (ak;bk], we can
establish nite additivity of X: Let A = Sk(ak;bk] where (ak;bk] (0;1] are pairwise
disjoint. Then,
XA =
X
k
X(ak;bk].
Since Xt is Brownian motion, Xt has orthogonal increments,
kXAk2 =
X
k
kXbk Xakk2.
Furthermore, since our image space is L2( ;F;P),
kXb Xak2 = EjXb Xaj2 = EjXbj2 EjXaj2.
Then F(t) := EjXtj2 is a bounded nondecreasing function on [0;1], which then generates
a bounded Borel measure on B0. And so, we have for all A2B0 where A is the nite
union of pairwise disjoint intervals (ak;bk]
kXAk2 =
X
k
kXbk Xakk2 =
X
k
(ak;bk] = (A).
Now, let A 2B, and choose a collection An 2B0 such that (A An) ! 0. Then,
XAn is Cauchy in L2( ;F;P) and hence converges. De ne X : B ! L2( ;F;P) by
XA := limn!1XAn where An 2B0 and (A An) ! 0. It follows that if An 2B are
pairwise disjoint, then X(
[
n
An) =
X
n
X(An) a.e., and hence X is a countably additive
vector measure. This vector measure (or wide-sense random measure) generated by Brownian
motion is called white noise. From real analysis, we know that if a function is of bounded
variation, then its derivative exists almost everywhere. Therefore, since Brownian motion is
(almost surely) nowhere di erentiable, X is not of bounded variation.
16
The other sense of \random measure" is much narrower. Consider a countably-additive
vector-valued function F from a measurable space ( ;F) to an F-space, X L0(S;S;P),
where (S;S;P) is a probability space. This F is then a vector measure and a random measure
in the wide sense, but the narrow sense requires more.
De nition 1.28. A random measure is a kernel from a probability space (S;S;P) to a
measurable space ( ;F), that is, a function : S F!R+ such that ( ;E) is a random
variable on (S;S;P)16 and (s; ) is a measure on F P-a.s..
A random measure is often denoted by s(E) for all s2S and E2F, and the subscript
is often dropped. By using to refer speci cally to the mapping from F into L0(S;S;P)
given by (E) = ( ;E), we can refer to as a vector measure (provided, of course, that it
is additive). Then we would say a random measure is a function : ( ;F) !L0(S;S;P)
such that (E)2L0(S;S;P) for all E2F and s is a scalar measure on F for each s2S.
This notation also allows us to refer to as the family of random variables f E : E2Fg.
Notice that a random measure in the narrow sense is not, by de nition, additive. How-
ever, if, for any two disjoint measurable sets E0 and E1 in B, (E0 [E1;!) = (E0;!) +
(E1;!) a.e. for any ! 2 , then is a vector measure. If, for any collection (Ek)k2N of
pairwise disjoint open sets in F, (SkEk;!) = Pk (Ek;!) a.e. for any !2 , we say that
is a countably additive vector measure. Because is originally de ned as taking values
in R+, we can see that there is a direct correspondence in whether ( ;E) is ( -) nite and
whether (!;E) is ( -) nite.
For the sake of an example, we will progress towards a Poisson process via random
measures and point processes, a move which will be facilitated by the following theorem
from T.E. Harris [9]:17
Theorem 1.29 (Harris). Let S be a (complete) metric space, B the Borel - eld of S, and
X =fXB : B2Bg a random process on B such that
16That is, an element of L0(S;S;P).
17This is a version of the theorem from O. Kallenberg. The original statement of the theorem can be found
in Appendix F.
17
1. XB 0 for all B2B,
2. XB1[B2 = XB1 +XB2 a.s. for all pairs B1 and B2 of disjoint sets in B, and
3. XBnP !0 as Bn!;.
Then there exists a random measure on S such that B = XB a.s. for all B2B.
De nition 1.30. A random measure on a measurable space has independent incre-
ments if for any disjoint sets E1;:::;En2F, the random variables E1;:::; En are indepen-
dent.
De nition 1.31. A point process on a space S is a locally nite random measure from
(S;S;P) to (Rd;Bd) that takes values in Z for all bounded B2Bd.
A point process can be thought of as a count of a collection of random points.
Example 1.32. A classic example of a point process is a \counting process". Take a random
sequence (Xn) with values in Rd and let E be a count of Xn?s in E, i.e. E =
X
n
1IE(Xn).
With these two de nitions in mind, we can see that a standard Poisson process is an
example of a random measure.
De nition 1.33. A Poisson process on a measurable space ( ;F) with intensity measure
, is a point process on with independent \increments" (i.e. disjoint sets) such that E
has Poisson distribution with mean (E) for all E2F such that (E) <1.
If we have the luxury of having our point process on (R+;B; ), our points will have a
linear ordering, allowing us to discuss increments.
De nition 1.34. A Poisson process with rate is a family of random variables Nt, t 0
such that
1. if 0 = t0 s> 0.
As mentioned before, may be considered as the family of random variablesf E : E2
Fg. Hence, this is a speci c case where = [0;1) and F = B. (Speci cally, N(tk)
N(tk 1) = (tk 1;tk]). Furthermore, here is given by (s;t] = (t s).
As de ned, a Poisson process must be a random measure. However, even without this
progression of de nitions, it is not very suprising that a \counting" process would be a
random measure:
Let (S;S; ) be a - nite separable measure space and ( ;F;P) a probability space.
Let Xn : !S be random elements (i.e. measurable functions), and let a full event (i.e. an
event with full measure) 0 be the common domain of the sequence Xn. De ne the counting
measure as
(NS)(!) =
X
n
1IS(Xn(!)), for a given S2S and all !2 0.
Then, for all ! 2 0, it is the counting measure of the sequence (xn) = (Xn(!)).
However, we also have de ned a vector measure with values in L2( ;F;P):
S7!N(S)( ) =
X
n
1IS(Xn( )) for all S2S.
There do, however, exist vector measures of the form F : ( ;F) !L0(S;S;P) such
that F(E; )2L0(S;S;P) for all E2F and F( ;!) is not a scalar measure on F for each
!2 .
Example 1.35. Let (rn) be a sequence of Rademacher functions18 de ned on ([0;1];B;P)
and (an)2?2(R)n?1(R). De ne the Rademacher measure on (N;2N;c) as the vector measure
with values in L2([0;1];B;P) by
F(K) =
X
n2K
anrn, for all K22N.
18de ned in Appendix B
19
If c(K) =1, then the series converges in the metric space L2. Hence, we immediately
have unconditional convergence required for F to be a vector measure.
Then, for each in nite K N, there is a full event K such that for all !2 K,
X
n2K
anrn(!) = lim
N!1
NX
n=1
1In2Kanrn(!).
Now, to have a scalar measure a.s. on 2N when some ! 2 N is xed, we need, in
particular, that
F(N)(!) =
X
n2N
F(fng)(!) =
X
n2N
anrn(!) <1.
Since the collection (fng)n 2N is invariant under permutations,
X
n2N
anrn(!) must be
unconditionally convergent. In R, this is equivalent to absolute convergence. However,
X
n2N
janrn(!)j=
X
n2N
janj where (an) =2?1.
Therefore, F is a vector measure, but not a random measure in the narrow sense.
An analogous argument shows that Brownian Motion is not a random measure in the
narrow sense. (As previously established, it is a countable vector measure.) The claim that
BM is a random measure amounts to the claim that almost all paths of a given BM have
bounded variation on [0;1].
1.7 Conclusions
Now, we are equipped with a basic introduction to vector measures. In the next chapter,
we will explore expressing vector measures as integrals of Banach-valued functions (and vice
versa), which, in cases of a general domain space, requires some form of a Radon-Nikod ym
Theorem.
20
Chapter 2
Bochner Integration and the RNT
This section will develop a vector-valued integral of a vector-valued function with respect
to a scalar measure, called the Bochner integral. The Bochner integral will allow us to expand
the class of Lebesgue integrable functions to include vector-valued functions and to integrate
this new class of functions, which we will call Bochner integration. By de ning this vector-
valued integral and extending the class of Lebesgue integrable functions, we will be able to
formulate a Radon Nikod ym Theorem in the more general context of Banach spaces. Since
the theorem fails in some Banach spaces, the Radon Nikod ym property emerges.
For this and subsequent sections, we will consider a probability space ( ; ; ) and
Banach space X with underlying eld R. Again, all vector measures are understood to be
countably additive. There are analogous results for C, and many of these results extend to
other scalar elds and - nite spaces, but since the aim is to understand concepts applying to
functions between probability spaces and Banach spaces, we will not obscure these concepts
with other scalar elds.
2.1 Bochner Integral for -Simple Functions
De nition 2.1. A function f : !X is -simple if f(!) =
nX
i=1
1IEixi where E1;:::;En are
disjoint1 members of , (Ei) <1 for all i, and x1;:::;xn2X. 2
We need not require that (Ei) <1for all i, but this requirement will allow our results
to extend to - nite measure spaces, since it will force the integrals of our simple functions
to be well-de ned. Now, we will de ne the Bochner integral for -simple functions.
1They need not be disjoint yet, but WLOG, it?s convenient to make this assumption at the outset.
2Of course, it is implicit that f : ( Sn
i=1Ei)!f0g X.
21
De nition 2.2. Let f : !X is -simple; then for any E2 , its Bochner integral is
de ned as
Z
E
f d =
nX
i=1
(E\Ei)xi
where f(!) =
nX
i=1
1IEixi with E1;:::;En disjoint members of and x1;:::;xn2X.
Proposition 2.3. The Bochner integral of a -simple function f is independent of the
representation of f.
Proof. Let f(!) =
nX
i=1
1IEixi =
mX
j=1
1IAjyj on a measurable set E where E1;:::;En and
A1;:::;Am are collections of disjoint measurable sets,
n[
i=1
Ei =
m[
k=1
Aj = E, and
x1;:::;xn;y1;:::;ym 2 X. Then consider the re nement of both partitions, fFi;jg where
Fi;j = Ei\Aj for all i;j. Then, f(!) =
nX
i=1
mX
j=1
1IFi;jxi =
nX
i=1
mX
j=1
1IFi;jyj. Furthermore,
nX
i=1
(E\Ei)xi =
nX
i=1
mX
j=1
(E\Fi;j)xi and
mX
j=1
(E\Aj)yj =
nX
i=1
mX
j=1
(E\Fi;j)yj
Thus, it remains to show that
nX
i=1
mX
j=1
(E\Fi;j)xi =
nX
i=1
mX
j=1
(E\Fi;j)yj, which follows
from the observation that, for each pair i;j, given Fi;j6=;, we have that xi = yj.
Before we continue, we will establish linearity, additivity, and contractivity for Bochner
integrals of -simple functions.
Proposition 2.4 (Linearity). Let f;g : !X be -simple functions and 2R. Then, for
all E2 ,
1.
Z
E
f d =
Z
E
f d
2.
Z
E
f +g d =
Z
E
f d +
Z
E
g d
22
Proof. Since the claim for E will be analogous, we will show the claim for E = . Also,
say f =
nX
i=1
1IBixi and g =
mX
j=1
1IAjyj where x1;:::;xn;y1;:::;yn 2X and (Bi) and (Ai) are
nite collections of disjoint measurable sets.
1. By the properties of vector spaces,
Z
f d =
nX
i=1
(Bi) xi =
nX
i=1
(Bi)xi =
Z
f d .
2. Let Ei;j = Bi\Aj for all i;j. Then (Ei;j) is a re nement of both (Bi) and (Aj), and
we can de ne the simple function f +g as
nX
i=1
mX
j=1
1IEi;j(xi +yj). Then,
Z
f d +
Z
g d =
nX
i=1
(Bi)xi +
mX
j 1
(Aj)yj =
nX
i=1
mX
j=1
(Bi\Aj)(xi +yj)
=
nX
i=1
mX
j=1
(Ei;j)(xi +yj) =
Z
f +g d
Corollary 2.5 (Additivity). If (Ei)ni=1 are disjoint measurable sets whose union is E2
and f : !X is a -simple function then
nX
i=1
Z
Ei
f d =
Z
E
f d .
If we let fi = f 1IEi, the proof follows from linearity.
Proposition 2.6 (Contractivity). Let f : !X be a -simple function. Then
Z
E
f d
Z
E
kfkd .
Proof. WLOG, we will show the claim for E = . Let f =
nX
i=1
1IEixi. Then,
Z
f d
=
nX
i=1
(Ei)xi
nX
i=1
(Ei)kxik=
nX
i=1
Z
Ei
kxikd =
Z
kfkd
23
Example 2.7. For n 2N and 1 i n, let gi;n : [0;1] !R be the constant function
gi;n(x) = in. Then, we can de ne the -simple functions fn : ([0;1];B; )!L1[0;1] by
fn(x) =
nX
i=1
1I(i 1
n ;
i
n]
gi;n(x)
!
. 3 Then for each n2N,
Z
fn d =
nX
i=1
i 1
n ;
i
n
gi;n(x)
!
=
nX
i=1
1
ngi;n(x).
Since gi;n(x) is the constant function gi;n(x) = in, for each i;n, 1ngi;n(x) is the constant
function g0i;n(x) = in2 . Then
nX
i=1
1
ngi;n(x) =
nX
i=1
g0i;n(x) = Gn(x) where Gn(x) is the constant
function Gn(x) = n+ 12n .
2.2 Bochner integral de ned
Now, we will de ne the Bochner integral for more general functions, but rst, we must
de ne what it means for a function to be (strongly) -measurable.
De nition 2.8. f : !Xis strongly -measurable if there exists a sequence of -simple
functions (fn) where limn!1kfn fk= 0 -a.e..
Henceforth, we shall just call such a strongly -measurable function -measurable. (We
will leave the discussion of weakly measurable function to section 2.4.)
Next, we de ne Bochner integrability for a -measurable function.
De nition 2.9. Let f : !X be strongly -measurable; f is Bochner integrable if
there exists a sequence of -simple functions (fn) that converge to f -a.e. and
limn!1
Z
kfn(!) f(!)kd (!) = 0.
Observe that, once we compose the norm ofXwithfn f, we have a real-valued function,
and once we have a real-valued (Lebesgue integrable) function, the Bochner and Lebesgue
3Where fn(0) = 0 1If0g.
24
integrals are the same. Hence, Bochner integration can be thought of as generalizing the
class of Lebesgue integrable functions.
There is one exception to their coincidence, however. The Lebesgue integral (of a non-
integrable function) can be in nity, whereas a in nity would have to be formally de ned (and
R extended to R) for a Bochner integral of a real-valued function to yield in nity. (In other
words, the Lebesgue integral can be de ned for a function that is not Lebesgue integrable,
whereas the same makes no sense for the Bochner integral.)
De nition 2.10. Given that a function f : !X is Bochner integrable, the Bochner
integral of f is de ned for each E2 as
Z
E
f d = limn!1
Z
E
fn d
where (fn) is a sequence of -simple functions such that limn!1
Z
kfn(!) f(!)kd (!) = 0.
Proposition 2.11. The Bochner integral is well-de ned, i.e.
1. Consistency: The Bochner integral of f is independent of the sequence of simple func-
tions (fn) such that limn!1
Z
kfn(!) f(!)kd (!) = 0.
2. Uniqueness: If f : !X is Bochner integrable and (fn) is a sequence of -simple
functions such that limn!1
Z
kfn(!) f(!)kd (!) = 0, then limn!1
Z
E
fn d is unique.
Proof. 1. Consistency: Suppose there exist two sequences of simple functions (fn) and
(gn), each of which converge to f a.e. and
limn!1
Z
kfn(!) f(!)kd (!) = limn!1
Z
kgn(!) f(!)kd (!) = 0.
WLOG, we will show the claim for E = . Let > 0. Then there is a N
0 such that for all n N,
Z
kfn fk d < =4,
Z
kgn fk d < =4; and
Z
gn d limn!1
Z
gn d
< =2: Then for all n N,
Z
kgn fnkd
Z
kgn fkd +
Z
kf fnkd < =2.
25
Hence,
=2 >
Z
kgn fnkd
Z
gn fn d
=
Z
gn d
Z
fn d
.
Then for all n N,
Z
fn d limn!1
Z
gn d
Z
fn d
Z
gn d
+
Z
gn d limn!1
Z
gn d
< .
2. Uniqueness: WLOG, we will show the claim for E = . Let (fn) a sequence of -simple
functions from to X that converges to f : !X -a.e. such that limn!1
Z
kfn
fkd = 0. Suppose (gm) is also a sequence of -simple functions from to X that
converges to f : !X -a.e. such that limn!1
Z
kgm fkd = 0. Let > 0. Then,
there exists an N0 0 for which
Z
gm d
Z
fn d
=
Z
gm fn d
Z
kgm fnkd
Z
kgm fkd +
Z
kf fnkd <
Similarly, there is an N1 0 for which for all n;m N,
Z
gn d
Z
gm d
< and
Z
fn d
Z
fm d
< .
Hence, if we de ne the sequence (hk) in X such that h2i 1 =
Z
gi d and h2i =
Z
fi d . Then there exists an N 0 (N = maxfN0;N1g) such that for all n;m N,
khn hmk < . Hence (hk) is Cauchy and thus converges since X is complete as a
Banach space. Since
Z
gm d
and
Z
fn d
are Cauchy subsequences of (hk),
they converge to the same limit.
Next, we establish contractivity for Bochner integrable functions.
Proposition 2.12 (Contractivity). If f is Bochner integrable, then
Z
f d
Z
kfkd .
26
Proof. Since f is Bochner integrable.
Z
f d
=
lim
n!1
Z
fn d
= lim
n!1
Z
fn d
lim
n!1
Z
kfnkd
Let > 0. Then there exists an N > 0 such that for all n N,
>
Z
kf fnkd
Z
kfkd
Z
kfnkd
.
Hence limn!1
Z
kfnkd =
Z
kfkd , and, therefore,
Z
f d
Z
kfkd .
Furthermore, by a classic diagonalization argument, we can now go beyond simple func-
tions converging to f:
Corollary 2.13. Suppose f : !X is -measurable and (fn) is a sequence of Bochner
integrable functions that converge to f -a.e. such that limn!1
Z
kfn fk d = 0. Then
Z
f d = limn!1
Z
fn d .
2.3 Bochner?s Characterization
In this section, we will give an essential characterization of Bochner integrable functions.
The result is attributed to Bochner.
Theorem 2.14 (Bochner?s Characterization). Let f : !X be a -measurable function,
then f is Bochner integrable if and only if kfk is Lebesgue integrable.
Proof. )Suppose f is Bochner integrable and (fn) is a sequence of -simple functions such
that lim
n!1
Z
kfn fkd = 0. Then for any > 0, there exists an N > 0 such that for all
n;m N,
kfn fmkL1(X) =
Z
kfn fmkd
Z
kfn fkd +
Z
kf fmkd < .
In other words, (fn) is Cauchy in L1(X), which is a metric space. Therefore, (fn) is bounded
in L1(X). Furthermore, since for any > 0 there exists an N > 0 such that for all n N,
>
Z
kfn fkd
Z
kfnkd
Z
kfkd
, we know that lim
n!1kfnkL
1(X) = kfkL1(X).
27
Therefore, since (fn) is bounded in L1(X) and converges to f in L1(X), kfkL1 < 1, i.e.
kfk2L1(X).
( Suppose
Z
kfkd <1.
Let (gn) be a sequence of -simple functions that converge to f -a.e.. (Then, by the
triangle inequality, kgnk!kfk -a.e..) De ne fn := 1Ikgnk 2kfkgn.4 Then (fn) is a sequence
of -simple functions that converge to f -a.e.. Furthermore, since kfnk 2kfk for all n
and
Z
2kfkd <1, by the Dominated Convergence Theorem, lim
n!1
Z
kfn fkd = 0.
A natural question is whether or not Bochner?s characterization will still hold if ( ; ; )
is a - nite measure space. This is where our requirement that (Ei) <1(where the Ei are
the measurable sets used to de ne the -simple functions in De nition 2.1) comes in handy.
In fact, the same proof holds for - nite spaces. 5
With Bochner?s characterization, we will be able to use theorems already available to
Lebesgue integration to prove corresponding theorems for Bochner integration.
Proposition 2.15. Suppose f;g Bochner integrable and 2R, then
1.
Z
f d =
Z
d d and
2.
Z
f +g d =
Z
f d +
Z
g d
Proof. Let (fn) and (gn) be sequences of -simple functions that converge -a.e. to f and g
respectively such that
Z
f d = limn!1
Z
fn d and
Z
g d = limn!1
Z
gn d .
1. By Bochner?s characterization, f is Bochner integrable. Then, since ( fn) is a se-
quence of -simple functions that converges -a.e. to f and limn!1
Z
k fn fkd = 0,
we have that
4The choice of the fn comes from [13].
5Without the stipulation that (Ei) < 1 for all i in the de nition of -simple functions, a simple
counterexample would be a R-valued simple function f = 1I[0;1) where
Z
f d =1.
28
Z
f d = limn!1
Z
fn d = limn!1
Z
fn d =
Z
f d .
2. Since f and g are Bochner integrable,
Z
kf + gkd
Z
kfkd +
Z
kgkd <1,
and therefore, f + g is Bochner integrable by Bochner?s characterization. Also, since
fn!f and gn!g -a.e., fn +gn!f +g -a.e.
Similarly, limn!1
Z
k(fn +gn) (f +g)kd = 0; thus
Z
f +g d = limn!1
Z
fn +gn d .
Hence, it will su ce to show that
Z
f d +
Z
g d = limn!1
Z
fn +gn d .
Let > 0. Then there exists an N 0 such that for all n N,
Z
fn +gn d
Z
f d +
Z
g d
=
Z
fn d +
Z
gn d
Z
f d
Z
g d
Z
fn d
Z
f d
+
Z
gn d
Z
g d
< :
Example 2.16. Consider the Banach space C[0;1] (i.e. the space of all R-valued contin-
uous functions on the compact set [0;1]) and the measure space (N;2N;c) where c is the
counting measure, and we will construct not just one, but a sequence of simple functions
BN : (N;2N;c)!C[0;1]. 6
6Yes, (N;2N;c) is not only not a probability space, but is not even nite. However, since Bochner?s
characterization holds in - nite spaces, we will avail ourselves of this fact in order to provide a particularly
interesting example.
29
First, let T be the tent map on [0;1] with height 1.
0 1
1
Figure 2.1: T : [0;1]!R
Next, extend T periodically to R (with period 1), and call this new map b(x).
0 1
1
Figure 2.2: b : R!R
Then, for all n2N, de ne bn(x) = b(2
nx)
2n [0;1]
b1(x)
1
1 b
2(x)
1
1
Figure 2.3: b1 : [0;1]!R and b2 : [0;1]!R
Then bn2C[0;1] and kbnk= sup
x2[0;1]
bn(x) = 12n for all n.
De ne a sequence of simple functions Fm : (N;2N;c) ! C[0;1] by Fm =
mX
n=1
bn1Ifng.
Then
Z
kFmkdc =
mX
n=1
2 n <1. Let F : (N;2N;c) !C[0;1] be given by F =
X
n
bn1Ifng.
Then Fm!F a.e., and kFk=
X
n
kbnk1Ifng =
X
n
1
2n1Ifng. Hence,
30
Z
N
kFkdc =
Z
N
X
n
1
2n1Ifng dc =
X
n
1
2nc(fng) =
X
n
1
2n <1
Therefore, F is Bochner integrable, and furthermore,
Z
N
F dc = lim
k!1
Z
N
kX
n=1
bn1Ifng dc = lim
k!1
kX
n=1
bnc(fng) = lim
k!1
kX
n=1
bn =
1X
n=1
bn
In fact,
Z
N
F dc is a continuous nowhere di erentiable function called a Bolzano function.
2.4 Pettis Integral
Before we continue, we will digress brie y to at least mention the more general Pettis
integral. First, we need to de ne weak -measurability.
De nition 2.17. A function a f : !X is weakly (or scalarly) -measurable if x f
is -measurable for each x 2X .
Proposition 2.18. A (strongly) -measurable function is weakly -measurable.
Proof. Suppose f is strongly -measurable, and (fn) is a sequence of -simple functions such
that limn!1kfn fk= 0, -a.e. Let x 2X . Then for each fn,
x (fn) = x
k
nX
i=1
xin1IEin
!
=
knX
i=1
x (xin)1IEin
is -simple. Furthermore,
lim
n!1
kx (fn) x (f)k= lim
n!1
kx (fn f)k lim
n!1
kx k kfn fk= 0
Bochner integral theory does not apply directly to functions that are weakly -measurable
or whose norm is not Lebesgue integrable. Herein lies one of the merits of the Pettis integral.
But rst, a lemma.
Lemma 2.19 (Dunford). [2] Suppose f : !X is a weakly -measurable function and
x f 2L1( ) for each x 2X . Then for each E2 , there exsits ?E 2X such that, for
all x 2X ,
31
?E(x ) =
Z
E
x (f) d
De nition 2.20. [2] If f is a weakly -measurable X-valued function on such that x f2
L1( ) for all x 2X , then f is called Dunford integrable. The Dunford integral of f
is the functional ?2X such that, for all x 2X ,
?(x ) =
Z
x (f) d .
Notice that x (f) takes values in the underlying scalar eld. Hence, the integral,Z
x (f) d , is the Lebesgue integral, and so, for any E2 , we can de ne
Z
E
x (f) d :=
Z
x (f)jE d =
Z
x (fjE) d .
For each E 2 ; we call the functional ?E 2X given by ?E(x ) =
Z
E
x (f) d for all
x 2X the Dunford integral of f over E.
If ?E 2X7 for each E2 , then f is called Pettis integrable, and ?E is the Pettis
integral of f over E. If X is a re exive, then Dunford integrability is Pettis integrability.
Now, consider a function f : ! X that is -measurable (and hence weakly -
measurable) and Bochner integrable. By Bochner?s characterization, that means
Z
kfkd <
1. Let x 2X . Then,
Z
jx (f)jd kx k
Z
kfkd <1.
Hence, Bochner integrability implies Dunford integrability.
Example 2.21. Let X be a Banach space, and (xn)2XN be summable. De ne f : N!X
by f(n) = xn for each n2N. Then, f is measurable (and hence -weakly measurable).
To see that f is Dunford integrable, let x 2X . Then,
Z
jx (f)jdc =
Z
x
X
n2N
xn1Ifng
!
dc =
Z
X
n2N
x (xn)1Ifng
dc
Z X
n2N
jx (xn)j1Ifng dc =
X
n2N
jx (xn)j:
7Of course, by E 2X, we mean E 2J(X) X where J : X!X is the natural embedding where
for each x2X and x 2X ,Jx(x ) = x (x).
32
Since x is continuous, (x (xn)) is summable in R, which means it is also absolutely conver-
gent. Hence
X
n2N
jx (xn)j<1, and f is Dunford integrable.
The Dunford integral for each E N is the functional E = X given by
E(x ) =
Z
E
x
X
n2N
xn1Ifng
!
dc =
Z X
n2N
x (xn)1Ifng dc =
X
n2E
x (xn)
for all x 2X . In bracket notation, E =
X
n2E
h ;xni = h ;
X
n2E
xni. In other words, f is
Pettis integrable.
As for Bochner integrability, by Bochner?s characterization, f2L1(X;c) i kfk2L1(c)
i
Z
kfkdc =
X
n2N
kxnk<1. However, from Orlicz?s Theorem (in Chapter 1), we know that
summability and absolute convergence are the same i X is a nite-dimensional Euclidean
space. 8 Hence, if X is nite-dimensional, then f is Dunford integrable i f is Bochner
integrable, and if X is in nite-dimensional, then we have only that Bochner integrability of
f implies Dunford integrability of f.
However, even in this discrete space there are Dunford integrable functions that are not
Pettis integrable.
Example 2.22. Consider X = c0 and f : N! c0 given by f(n) = en where (en) is the
standard basis for c0. Let x 2c 0, x = (xn)2?1 such that x =hx; i, and E22N. Then,
Z
E
jx (f)jdc =
X
n2E
jxnj<1.
However, for any x 2c 0 with x = (xn)2?1 such that x =hx; i and E22N,
Z
E
x (f) dc =
X
n2E
xn =hx;(1)1n=1i.
But, (1)1n=1 =2c0.
This example extends readily to one de ned on a measurable space that is not discrete.
8The \only if" portion comes from [5].
33
Example 2.23. Considerc0 with standard basis (en), and let our measure space be ([0;1];B; )9
De ne fn : [0;1] !c0 by fn := nen1I(0;1
n]
for all n2N where (en) is the standard basis for
c0, and de ne f : [0;1]!c0 by f :=
X
n2N
fn. To see that f is Dunford integrable, let x 2c 0
and x = (xn)2?1 such that x =hx; i. Then,
Z
jx (f)jd =
Z
jx
X
n2N
nen1I(0;1
n]
jd
Z X
n2N
njx (en)j1I(0;1
n]
d
=
X
n2N
njxnj1n =
X
n2N
jxnj<1
However, ? =2c0. To see this, let x 2c 0 and x = (xn)2?1 such that x =hx; i. Then,
Z
x (f) d =
Z
x
X
n2N
nen1I(0;1
n]
d =
Z X
n2N
n(x (en))1I(0;1
n]
d
=
X
n2N
n(xn) 1n =
X
n2N
xn
However,
X
n2N
xn =hx;(1)1n=1i where (1)1n=1 =2c0.
2.5 Extending R results
Several important results still hold even after we leave R for more general spaces, aside
from the ones already mentioned. In this section we will discuss a few that do extend and
why others do not. First, we will note that not all Banach spaces have a partial ordering.
Hence many theorems requiring or guaranteeing an inequality will no longer hold in all
Banach spaces, e.g. Monotonicity, Fatou?s Lemma, and the Monotone Convergence Theorem.
Theorems with non-negative assumptions also may not extend. However, we do have an
analogue to Lebesgue?s Dominated Convergence Theorem.
Theorem 2.24 (Dominated Convergence Theorem). [2] Let fn : !X be a sequence of
Bochner integrable functions, g : !R, g2L1( ), f : !X is a -measurable function
9where is the Lebesgue measure
34
such that (fn) converges to f in measure, andkfnk jgj -a.e. for all n. Then f is Bochner
integrable and limn!1
Z
E
fn d =
Z
E
f d for all E2 , and limn!1
Z
kf fnkd = 0.
Notice that, according to proceeding prepositions, all that we need to show is that
limn!1
Z
kfn fkd = 0. Also, note that a sequence of functions converges in ( nite) measure
i every subsequence contains an almost everywhere convergent sub-subsequence (all of which
converge to the same limit). Since we are working in a metric space, this subsequence clause
is just -a.e. convergence. Therefore, since we are working with functions from a nite
measure space to a metric space, we can just assume that (fn) ! f -a.e.. (If we were
working in a - nite space, a slightly weaker version of the DCT would hold, i.e. where
(fn)!f -a.e..) 10
Proof. Since (fn) converge to f -a.e., kfnkconverges to kfk -a.e., and limn!1kfn fk= 0.
Furthermore, sincekfnk g for all n,kfk g. Hence,kfn fk kfnk+kfk j2gj -a.e..
Then, by the classic Dominated Convergence Theorem,
limn!1
Z
kfn fkd =
Z
limn!1kfn fkd =
Z
0 d = 0.
We also get a generalized case of Egoro ?s Theorem:
Theorem 2.25. For a - nite measure space (S;S; ), if E 2S, (E) <1, and (fn) a
sequence of measurable functions on E, each of which is nite a.e. in E, that converges a.e.
to a nite -measurable f, then for all > 0, there is an A E such that (E A ) <
and (fn) converges to f uniformly on A .
The proof for this theorem is simply the standard proof for Egoro ?s Theorem with a
general norm in place of absolute value.
10Notice that Bochner?s characterization is not required for the Dominated Convergence Theorem. In fact,
given the relationship between Bochner and Lebesgue integration and the Dominated Convergence Theorem
for Lebesgue integration, the Dominated Convergence Theorem for Bochner integrals is more of a corollary
than a theorem.
35
Theorem 2.26. [2] If f is Bochner integrable with respect to , then
1. lim
(E)!0
Z
E
f d = 0,
2. (Countable Additivity) If (En) is a sequence of pairwise disjoint members of and
E =
1[
n=1
En, then
Z
E
f d =
1X
n=1
Z
En
f d ,
where the sum is absolutely convergent, and
3. If F(E) =
Z
E
f d , then F is of bounded variation and jFj(E) =
Z
E
kfkd for all
E2 .
Proof. 1. Since lim
(E)!0
Z
E
kfk d = 0 for f 2 L1( ), and f is Bochner integrable i
kfk2L1( ),
0 = lim
(E)!0
Z
E
kfkd lim
(E)!0
Z
E
f d
=
lim
(E)!0
Z
E
f d
.
2. Note rst that
1X
n=1
Z
En
f d is dominated term by term by the nonnegative series
1X
n=1
Z
En
kfkd and that, since
1X
n=1
Z
En
kfkd
Z
kfkd <1, the series converges.
Thus
1X
n=1
Z
En
f d is absolutely convergent.
Note also that (by the niteness of ), limm!1
1[
n=m+1
En
!
= 0, and thus by the rst
property, limm!1
Z
S1
n=m+1En
f d
= 0. Then,
Z
E
f d
mX
n=1
Z
En
f d
=
Z
S1
n=m+1En
f d
= 0.
Hence,
Z
E
f d = limm!1
mX
n=1
Z
En
f d .
36
3. Let E2
: Let be a partition of E. Then
X
A2
kF(A)k=
X
A2
Z
A
f d
X
A2
Z
A
kfkd =
Z
E
kfkd .
Therefore, jFj(E)
Z
E
kfkd .
Furthermore, since f is Bochner integrable, jFj(E)
Z
E
kfkd <1, so F is of
bounded variation.
: [2] Let > 0 and choose ffng simple such that limn!1
Z
kf fnkd = 0. Choose
m su ciently large such that
Z
kf fmkd < 2 and a partition of E such
that
X
A2
Z
A
fm d
=
Z
E
kfmk d .11 Choose a re nement 0 of such that
jFj(E)
X
B2 0
Z
B
f d
<
2.
12 Then, we still have
Z
E
kfmkd =
X
B2 0
Z
B
fm d
.
Furthermore,
X
B2 0
Z
B
f d
Z
B
fm d
X
B2 0
Z
B
f d
Z
B
fm d
X
B2 0
Z
B
f d
Z
B
fm d
X
B2 0
Z
B
kf fmkd
=
Z
E
kf fmkd < 2:
Therefore,
jFj(E)
Z
E
kfmkd
=
jFj(E)
X
B2 0
Z
B
fm d
<
11We can choose such that A = Ai for
nX
i=1
1IAixi = fm on E.
Then,
X
A2
Z
A
fm d
= X
A2
k (A)xAk=
X
A2
(A)kxAk=
X
A2
Z
A
kfmkd =
Z
E
kfmkd .
12We can do so by the the de nition of jFj(E).
37
Since this holds for all su ciently largem,jFj(E) = limn!1
Z
E
kfnkd
Z
E
kfkd .13
Corollary 2.27. [2] If f and g are Bochner integrable and
Z
E
f d =
Z
E
g d for each
E2 , then f = g a.e. with respect to .
Proof. Let F(E) =
Z
E
f g d = 0. Then by (3), 0 = jFj(E) =
Z
E
kf gkd . Hence
kf gk= 0, which means, f = g a.e. .
However, the Bochner integral does have results with no nontrivial Lebesgue analogue,
such as the following property:
Theorem 2.28. [2] Let T be a bounded linear operator 14 If f is Bochner integrable with
respect to , then so is Tf and for all E2
T
Z
E
f d
=
Z
E
Tf d .
Proof. Since T is bounded,
Z
E
kTfk d
Z
E
Mkfk d < 1. Hence, Tf is Bochner
integrable. Also, the claim holds for f -simple by linearity.
Let (fn) be a sequence of -simple functions such that limn!1
Z
E
fn d =
Z
E
f d . Then
by the continuity of T,
T
Z
E
f d
= T
limn!1
Z
E
fn d
= limn!1T
Z
E
fn d
= limn!1
Z
E
Tfnd = limn!1
Z
E
Tf d
Finally, before we move forward, we should note a new class of spaces at our disposal:
De nition 2.29. For 1 p<1, de ne Lp( ;X) to be the vector space of all (equivalence
classes) of -measurable functions f : !X for which,
Z
kfkp d <1.
13by Fatou?s Lemma
14The theorem holds for closed linear operators as well, but we shall only require the result for bounded
linear operators. The more general theorem is attributed to Hille.
38
Much like the Lp spaces de ned with the Lebesgue integral, Lp( ;X), with the norm
k kLp( ;X) given bykfkLp( ;X) =
Z
kfkp d
1=p
for all f2Lp( ;X), is also a Banach space.
In particular, L1( ;X) is the space of all Bochner integrable functions from to X.
De nition 2.30. De ne L1( ;X) to be the vector space of all (equivalence classes) essen-
tially bounded -measurable functions f : !X
Again, L1( ;X), with the norm k kL1( ;X) given by
kfkL1( ;X) = inffM2R+ :kfk M a.e..g,
is also a Banach space.
2.6 The Radon Nikod ym Property
In our extended class of Lebesgue integrable functions, several results (such as the
Dominated Convergence Theorem) that held for the class of real-valued Lebesgue integrable
functions hold trivially (after a little reformulation) for the extended class. However, we shall
see that the Radon Nikod ym Theorem requires alteration before it holds in this larger class of
functions. Our focus in this section will be not in altering the theorem but in understanding
the signi cance of its failure to hold for measures taking values in certain Banach spaces.
First, let us put the classic Radon Nikod ym Theorem in the context of Banach spaces
and Bochner integrals:
Theorem 2.31 (Radon-Nikod ym Theorem). Assume ( ; ; ) is a nite measure space15
and X a Banach space. If F : !X is a -continuous vector measure of bounded variation,
then there exists a Bochner integrable g2L1( ;X) such that F(E) =
Z
E
g d for all E2 .
We call g the Radon-Nikod ym Derivative (RND) of F.
In the general context of Banach spaces and Bochner integrals, this theorem no longer
holds. The following is a classic incidence of the failure of the Radon-Nikod ym Theorem
from Diestel and Uhl:
15That is, a nite, countably additive set function on that takes values in [0;1).
39
Proposition 2.32. [2] There exists a countably additive co-valued vector measure of bounded
variation with no Radon-Nikod ym Derivative.
Proof. Let be the Lebesgue measure on ([0;1];B). ForE2B, write n(E) =
Z
E
sin(2n t)dt
and let F(E) = ( n(E))1n=1.16 Then F : B!c0 by the Riemann-Lebesgue Lemma17, and
for all E2B,
kF(E)kc0 sup
n
Z
E
jsin(2n t)jdt
Z
E
1 dt = (E).
Then, F is countably additive, -continuous, and of bounded variation.
1. To see that F is countably additive, let fEkg be a countable disjoint collection of
elements of B, and let E := SkEk. Then,
kF(E)kc0 =k( n(SkEk))nkc0 =k(Pk n(Ek))nkc0 1.
Hence, jPk n(Ek)j 1 for all n2N. If we let (en)1n=1 be the standard basis of c0,
then we have
F(E) = (
X
k
n(Ek))n =
X
n
X
k
n(Ek)
!
en =
X
k
X
n
n(Ek)en =
X
k
F(Ek)
2. To see that F is -continuous, rst note that, since F and are countably additive
and B is a -algebra, it will su ce to show that (E) = 0)F(E) = 0 for all E2B,
which follows immediately from the inequality.
3. To see that F is of bounded variation, let fEkgnk=1 be a pairwise disjoint collection in
B. Then,
nX
k=1
kF(Ek)kc0
nX
k=1
(Ek) 1. Hence, jFj([0;1]) 1 <1.
Now, suppose that F does have a RND, i.e. a Bochner integrable function f : [0;1] !c0
such that F(E) =
Z
E
f d for all E 2B. Then, f = (fn)1n=1 where, for each n, fn is the
16Note that sin(2n t) is merely the sine function with period 1
2n 1 .
17Speci cally, lim
k!1
Z 1
0
sin(kt) dt = 0, since E [0;1] and 2n !1, limn!1 n(E) = 0
40
nth coordinate functional on c0 for (en). Hence, each fn is bounded (continuous) and thus
measurable.
By Corollary 2.25, jFj(E) =
Z
E
kfkd for all E2B. Hence,
1
Z
E
kfk d =
Z
E
sup
n
jfnj d for all E 2B. It follows that
mX
n=1
fnen is Bochner
integrable for each m2N.18 Therefore, by the general Dominated Convergence Theorem,
for all E2B
F(E) =
Z
E
f d = lim
n!1
Z
E
mX
n=1
fnen d = lim
n!1
mX
n=1
Z
E
fn d
en =
Z
E
fn d
1
n=1
Therefore, by Corollary 2.26, fn(t) = sin(2n t) for almost all t2[0;1].
Now, let En = ft 2 [0;1] : fn(t) 1p2g. Then (En) = 14 for all n,19 and for all
t2lim sup
n
En, f(t) =2c0 since fn(t) 1p2 for all n. Also,
(lim sup
n
(En)) =
1\
k=1
1[
n=k
En
!
inf
k 1
1[
n=k
En
!!
inf
k 1
(sup
n k
(En)) 14
Therefore, ft2[0;1] : f(t)2c0g 34 < 1, contradicting f : [0;1]!c0.
Because this theorem may or may not hold in a given Banach space, we can think the
success or failure of the theorem to hold in the space as a property of the space itself, that
is, we have the following property:
De nition 2.33. A Banach space X has the Radon Nikod ym Property (RNP) with
respect to ( ; ; ) if for each -continuous vector measure, F : !Xof bounded variation,
there exisits a Bochner integrable f : !X such that F(E) =
Z
E
f d for all E2 .
De nition 2.34. A Banach space X has the Radon Nikod ym Property if it has the
Radon Nikod ym Property with respect to every nite measure space. 20
18Using Bochner?s Characterization,
Z
mX
n=1
fnen
d
Z mX
n=1
jfnjd <1
19Again, fn(t) is merely the sine function with period 1
2n 1 . Since sin(t) >
1p
2 for
1
4 of each period, it
follows that (En) = 14.
20In fact, it is enough that X has the RNP with respect to ([0;1];B; ).
41
Hence, from proposition 2.32, c0 does not have the Radon Nikod ym Property.
2.6.1 Signi cant Theorems
As we have noted, the RNP gives signi cant insight into analytic properties and the
geometry of a Banach space. To illustrate this, we shall brie y enumerate several signi cant
results pertaining to the RNP. The following theorems and proofs can be found in Diestel
and Uhl?s Vector Measures, [2].
Theorem 2.35 (Dunford-Pettis). Separable dual spaces have the Radon-Nikod ym Property.
Theorem 2.36 (Phillips). Re exive Banach spaces have the Radon-Nikod ym Property.
Theorem 2.37 (Uhl). If every separable closed linear subspace of X is isomorphic to a
subspace of a separable dual space, then X has the Radon-Nikod ym Property.
Theorem 2.38 (Von Neumann). Hilbert spaces have the Radon-Nikod ym Property.
Theorem 2.39. Let ( ; ; ) be a nite measure space and X a Banach space such that X
and X have the RNP. A subset K of L1( ;X) is relatively weakly compact if
1. K is bounded,
2. K is uniformly integrable, and
3. for each E2 , fREf d : f2Kg is relatively weakly compact.21
Theorem 2.40. For a Banach space X, the following are equivalent:
1. X has the RNP.
2. X has the Kre in-Mil?man property22.
3. Every separable subspace of X has a separable dual.
21In fact, without the RNP, we can guarantee a K that is not relatively weakly compact event if the three
criteria hold.
22That is, if each closed bounded convex subset of X is the norm closed convex hull of its extreme points
42
4. Ever separable subspace of X is a subspace of a separable dual space.
Theorem 2.41. If X is a weakly sequentially complete Banach space and X has the RNP,
then X is re exive.
Theorem 2.42 (Davis-Phelps). A Banach space has the RNP i each of its equivalent norms
has a dentable23 closed unit ball.
Theorem 2.43 (Hu -Morris). For a Banach space X, the following are equivalent:
1. X has the RNP.
2. Every closed bounded subset of X contains an extreme point24 of its closed convex hull.
3. Every closed bounded subset of X contains an extreme point of its convex hull.
4. For each closed bounded subset A of X, there is a nonzero x 2X and x0 2A such
that x (x0) = supx (A).
5. For each closed bounded subset A of X the collection of x 2 X that attain their
maxima on A is norm-dense in X .
Theorem 2.44. A Banach space lacks the RNP i there is a bounded open convex set K in
X and a norm closed subset A of K such that co(A) = K.
2.7 Radon Nikod ym Theorem for Bochner Integration
Thanks to the illuminary properties of the Radon Niko ym Property, the direct trans-
lation of the classic Radon Nikod ym Theorem into the context of Banach spaces is of most
interest. However, there exist several versions of the theorem that hold in Banach spaces,
and we will not leave this chapter without a brief discussion. The rst version is by Dunford
23A subspace D of a Banach space is dentable if for all > 0, there exists an x 2 D such that x =2
co(DnB (x)).
24An extreme point of a set is a point that is not an interior point of any line segment lying entirely within
the set{ a vertex, if you will.
43
and Pettis in 1940. In 1943, Phillips proved an extension of their result, then Metivier proved
that the converse of Phillips?s theorem holds. Finally, in 1968, Rie el proved an even more
extensive version.[11]
We shall not prove all of these here. In fact, this section will closely follow the paper
\Radon-Nikod ym Theorems for the Bochner and Pettis Integrals" by S. Moedomo and J.J.
Uhl, Jr.([11]). We will establish the necessary conditions for Rie el?s statement of the theo-
rem (as it appears in Moedomo and Uhl?s paper) and Phillips Theorem. Let ( ; ; ) be a
probability space and X a Banach space25.
Theorem 2.45 (Dunford-Pettis). [11] Let T : L1( ; ; )!X be a weakly compact operator
whose range is separable. Then there exists an essentially bounded -measurable g : !X
such that T(f) =
Z
fg d for all f2L1( ; ; ).
Theorem 2.46 (Phillips). [11] A vector measure F : !X is of the form F(E) =
Z
E
f d
for all E2 for some Bochner integrable f : !X if
1. F is -continuous,
2. F is of bounded variation, and
3. for each > 0 there exists an E 2 with ( E ) < such that for some weakly
compact A X, fF(E)= (E) : E E ; (E) > 0;E2 g A.
Theorem 2.47 (Rie el). [11]26 A vector measure F : !X is of the form F(E) =
Z
E
f d
for all E2 for some Bochner integrable f : !X i
1. F is -continuous,
2. F is of bounded variation, and
25Uhl and Moedomo assume nite, whereas Rie el assumes - nite, but we will stick with our probability
space from before.
26In his paper, ([16]), Rie el?s statement of the theorem includes two equivalent statements of the third
condition.
44
3. for each > 0 there exists an E 2 with ( E ) < such that for some compact
A X (with respect to the uniform operator topology on X),
fF(E)= (E) : E E ; (E) > 0;E2 g A .
Moedomo and Uhl?s proof of the necessary half of the claim is a simpli ed version of
Rie el?s proof that has been tweaked to show the claim holds (save the second part) for
Pettis integrable functions f. However, we shall stick to Bochner integrable functions. First,
we will show the necessary implication for Rie el?s Theorem. The rst two are satis ed by
Theorem 2.2627. Hence, all that remains is to prove (3).
Proof. Let F : !X be of the form F(E) =
Z
E
f d for all E 2 for some Bochner
integrable f : !X, and let (fn) be a sequence of -simple functions that converge a.e. to
f. Let > 0. Then, by Egoro ?s Theorem, there is an E 2 with ( E ) < such that
(fn) converges to f uniformly on E . Hence, since each fn is bounded, so is f on E . Then,
for all g2L1( ), de ne T;Tn : L1( )!X by T(g) =
Z
E
gf d and Tn(g) =
Z
E
gfn d for
all n. Since f is bounded and fn are bounded for all n, kgfk2L1( ) and kgfk2L1( ) for
all n. Hence, T;Tn2L(L1( );X) for all n.
Claim 1: limn!1Tn = T, i.e. kT Tnk= sup
kgk1 1
k(T Tn)gk!0.
Let g2L1( ) with kgk1 1. Then,
k(T Tn)gk=
Z
E
(gf gfn) d
Z
E
jgjkf fnkd
Z
E
jgjsup
!2E
kf(!) fn(!)kd
= sup
!2E
kf(!) fn(!)k
Z
E
jgjd
sup
!2E
kf(!) fn(!)k:
27(1) by (1) and (2) by (3)
45
Since (fn) converges to f uniformly on E , (Tn) converges to T on the uniform operator
topology.
Claim 2: For each n, Tn(L1( )) has nite dimension.
For each n, let fn =
knX
i=1
xi1IAi (where
kn[
i=1
Ai = E ). Then,
Tn(L1( )) =
Z
E
gfn d : g2L1( )
=
(Z
E
g
knX
i=1
xi1IAi d : g2L1( )
)
=
( k
nX
i=1
xi
Z
Ai
g
d : g2L1( )
)
=
( k
nX
i=1
Z
Ai
g d
xi : g2L1( )
)
Then for all x 2 Tn(L1( )), x =
knX
i=1
Z
Ai
g d
xi. Therefore, fx1;:::;xkng spans
Tn(L1( )), and hence dim(Tn(L1( ))) kn.
Therefore, each Tn is a compact operator. Since the collection of all compact operators
from L1( ) into X is closed in L(L1( );X), and since (Tn) converges to T in the uniform
operator topology, T is also a compact operator.
Now, consider the bounded set S = f1IE= (E) : E2 ; (E) > 0g in L1( ). For each
measurable E E , T(1IE) =
Z
E
1IEf d =
Z
E
f d = F(E). Therefore, the closure of
T(S) =fT(1IE= (E)) = 1 (E)
Z
E
f d = F(E) (E) : E2 ;E E g
is compact with respect to the uniform operator topology in X.
Now, we will use this result as well as the Dunford-Pettis Theorem to get Phillips
Theorem (i.e. the su ciency). But rst, a lemma:
Lemma 2.48. [11] A weakly compact operator T : L1( ; ; )!X has a separable range.
46
Proof. Let S = f1IE : E 2 g. Then the linear span of S is the collection of -simple
functions on , which is dense in L1( ; ; ). Therefore, by the linearity and continuity of
T, it will su ce to show that T(S) is separable.
Let f1IEng1n=1 2 SN and let 0 be the -algebra generated by f1IEng28. Then 0 is
countably generated and so L1( ; 0; ) = fg 2L1( ; ; ) : g 2 0g is separable. Since
T is continuous, T : L1( 0) !X is a weakly compact operator whose range is separable.
By Dunford-Pettis, there is an essentially bounded, 0-measurable f : !X for which
T(g) =
Z
gf d for all g2L1( ; ; ).
Now, let > 0, and kfk1 = M 2R+. Then, by the necessary direction of Rie el?s
Theorem, there is a set E 2 0 such that ( E ) < M+1 and a (norm) compact set
A X such that f
R
Ef d
(E) : E2 0;E E g A . Let A
0
=f x : 0 ( ) = 1 and
x2A g. Since A is compact, so is A0 . Now, note that
T(1IEn) =
Z
En
f d =
Z
En\E
f d +
Z
En E
f d .
Since En\E E , we have that
1
(En\E )
Z
En\E
f d 2A , and hence,
Z
En\E
f d 2 (En\E )A A0 .
Moreover,
Z
En E
f d
Z
En E
kfkd M (En E ) M ( E ) < .
Thus, T(1IEn) is within of a member of the compact set A0 . Then fT(1IEn)g is totally
bounded and hence is (norm) relatively compact. Thus fT(1IEn)g has a sequence that con-
verges in X. Then, for any in nite subset B T(S), there is a f1IEng T 1(B) such
that fT(1IEn)g has a convergent subsequence in X; hence B has a limit point in X. Then,
T(S) is relatively compact29. Since a compact subset of a metric space is separable, T(S) is
separable and hence so is T(S).
Now, we are ready to prove the main theorem.
28That is the smallest -algebra containing f1IE
ng.29
Because the previous property is equivalent to relative (or conditional) compactness
47
Theorem 2.49 (Phillips). [11] Let F : !X be a -continuous vector measure of bounded
variation such that for each > 0, there exists E 2 with ( E ) < such that
B =fF(E)= (E) : E E ;E2 ; (E) > 0g
is contained in a weakly compact subset of X. Then, there exists a -measurable Bochner
integrable function f : !X such that for E2 ,
F(E) =
Z
E
f d .30
Proof. For each n2Z+, choose En2 such that ( En) < 1n and
Bn :=fF(E)= (E) : E En;E2 ; (E) > 0g
is contained in a weakly compact An X. Let
S :=fg : !R : g =
nX
i=1
i1IEi; i2R;fEig are pairwise disjointg L1( ),
and de ne tn : S!X by31
tn
nX
i=1
i1IEi
!
=
nX
i=1
iF(Ei\En) =
X
i=1
i (Ei\En)F(Ei\En) (E
i\En)
The linearity of tn comes directly from the form of the elements of S.
Now, let S0 be the intersection of S and the unit ball in L1( ); hence S0 is a dense subset
of the unit ball of L1( ). Then for any f2S0, kfk1 1, and so
X
i=1
j i (Ei\En)j
nX
i=1
j ij(Ei) =kfk1 1.
Therefore,
tn(f)2co(Bn Bn) co(Bn Bn) = co(Bn Bn).
30In the paper, the authors also prove (without the requirement that F be of bounded variation) that F
can be given as the integral of a Pettis integrable function. However, for the sake of remaining conscise, we
will avoid the digression into Pettis integration.
31The last equality holds cleanly if we adopt the convention of 0=0 = 0.
48
Since An is weakly compact, so is An An; since Bn An, we have that Bn Bn An An.
Hence Bn Bn is weakly compact inX. Then, by the Krein-Smulian theorem32, co(Bn Bn)
is also weakly compact. Therefore, tn(S0) is contained in a weakly compact set. Since S0
is dense in the unit ball of L1( ), there is an extension Tn of tn on all of L1( ) that maps
the closed unit ball into a weakly compact set in X. Hence, Tn is weakly compact. By the
preceeding Lemma, Tn has a separable range, and hence by the Dunford-Pettis theorem,
there is a -measurable f : !X with support En33 such that Tn(g) =
Z
En
gfn d for all
g2L1( ).
If we do this for each n2Z+, we can produce an increasing sequence of measurable
sets (En) such that ( En)!0 and a sequence (fn) of -measurable Bochner integrable
functions such that
F(E\En) = Tn(1IE\En) =
Z
En
fn d .
Then fn1IEm = fm for n m since (En) is increasing. De ne f : !X by f :=
X
n2N
fn1IEn.
Since (En) % , fn !f uniformly, and hence f is -measurable, and fn = f1IEn for all
n2Z+.
Let E2 . Since F << , and limn ( En) = 0,
F(E) = limnF(E\En) = limn
Z
E\En
fn d = limn
Z
E\En
f1IEn d = limn
Z
E\En
f d
in the metric topology on X. Since F is of bounded variation,
1>jFj( )
Z
kfnkd =
Z
En
kfkd for all n2Z+.
Since kfnk%kfk, by the Monotone Convergence Theorem,
1>jFj( ) limn
Z
En
kfkd = lim
n
Z
kfnkd =
Z
kfkd .
Therefore, kfk is integrable, and hence f is Bochner integrable.
32The closed convex hull of a weakly compact subset of a Banach space is weakly compact
33That is, fn = fn1IE
n.
49
Therefore, (fn) !f in measure, kfnk kfk for all n where f is Bochner integrable;
hence, by the Dominated Convergence Theorem, limn
Z
E
fn d =
Z
E
f d for all E 2 .
Therefore, we have that f is Bochner integrable and
F(E) = limn
Z
E\En
f d = limn
Z
E
fn d =
Z
E
f d .
50
Chapter 3
Quantum Measures
Our nal chapter will explore an avenue of vector measure theory that branches from
what is called Quantum Probability. The operators discussed here are de ned and brie y
discussed in section D.4 of the Appendix.
3.1 Hilbert Space Quantum Mechanics
Among the contending axiomatic bases for quantum mechanics, the traditional approach
uses the structure of the Hilbert space and its self-adjoint operators to describe states and
observables within a given physical system. This approach was developed in the early 1930?s
by P. Dirac and J. von Neumann with the ideas of M. Born[8].
In both classical and quantum mechanics, a state represents a theoretically complete
description of a given physical system, and observables correspond to measurable quantities
in the physical system.[8] In the traditional axiomatic basis for quantum mechanics, the
physical system is represented by a complex Hilbert space, and states and observables are
described by entities in the Hilbert space. Hence the traditional axiomatic structure is
called Hilbert quantum mechanics. Von Neumann gave the following axioms for Hilbert
space quantum mechanics [8]:
(A1) The states of a quantum system are [described by] unit vectors in a complex Hilbert
space H.
(A2) The observables are [described by] self-adjoint operators on H.
51
(A3) The probability that an observable T has a value in a Borel set A Rwhen the system
is in the state is hPT(A) ; i where PT( ) is the resolution of identity (spectral
measure) for T.
(A4) If the state at time t = 0 is , then the state at time t is t = e itH=h , where H is
the energy observable and h is a constant (Planck?s constant).
In this chapter, we will focus on explicating A1 and A2. After which, we will introduce a
natural extension of those probabilities to vector measures on H, in the spirit of the classical
vector measure theory.
3.2 Basics of Quantum Probability
We begin the endeavor of de ning a probability with respect to our physical system,
represented by H.
3.2.1 Events
To de ne a probability, we will rst need to de ne events, which, in classic probability,
are elements of a -algebra on which the probability measure is de ned. We begin with the
following axiom:[8]
The events of a quantum system can be represented by projections on a
complex Hilbert space.
For the sake of restricting this discussion to merely one chapter, we will restrict ourselves to
bounded orthogonal projections on H. Because of our restriction, we now have a one-to-one
correspondence between events and closed subspaces of H, i.e.
Proposition 3.1. For each closed subspace E of H, there exists a unique orthogonal pro-
jection PE = projE that maps H onto E and is de ned such that for each x 2H, PE(x)
is the unique element in H such that (x PE(x)) 2E?.Furthermore, if P is a continuous
projection, then R(P) is closed and R(P) R(P)? = H.
52
1
Hence, the events for our probability are identi ed with closed subspaces of H. Further-
more, a subset E0 of H may be viewed as an event through its closed span:
E0 !spanE0 $PspanE0.
Example 3.2. To say a single vector h = (hj)2H is an event is to say that f h : 2Cg
or the projection Ph : H!H given by Ph = hh = [hjhk].
We will denote the family of events (i.e. bounded orthogonal projections onto closed
subspaces of H) by E and the \unit" by 1$H$PH.
3.2.2 Intersections and Unions
If we are to de ne a probability on elements of E, we must rst de ne unions and
intersections of \events" that are identi ed with bounded orthogonal projections.
The role of \intersection" will be played by product, i.e. composition. Here arise
some disanalogies between quantum and classic probabilities. First, the product is not
commutative in general, which leaves us with left and right \intersections". Second, E is
not necessarly closed under intersection. 2
There are two important exceptions to these two dis-analogies. First, if E F, then
PEPF = PE = PFPE. The second exception is of greater interest to us as it introduces the
notion of a disjoint union:
Proposition 3.3. If E and F are closed subspaces of H, then E and F are orthogonal i
PEPF =f0g= PFPE.
Proof. First, an intermediate claim: PFE =f0g i E F?.
Recall that every h2H can be written uniquely as h = x+y where x2F and y2F?,
and PFh = x. Then, y 2 F? () y = 0 + y where y 2 F? and 0 2 F is the unique
1As the proof of this proposition is standard and routine, we will omit it here.
2However, we need only the notion of disjoint events (which we will introduce momentarily) to talk about
a probability.
53
representation of y in H = F F?()PFy = 0. This proves our claim. Now,
PEPF = 0()PEPFx = 0 8x2H()PE(F) =f0g()F E?
()E and F are orthogonal ()E F?()PF(E) =f0g
()PFPEx = 0 8x2H()PFPEx = 0
De nition 3.4. We say two events PE and PF are disjoint if PEPF =f0g= PFPE.3
Intuitively, we say two events are disjoint if one?s occurance precludes the other?s oc-
curance and vice versa. Similarly, as observables correspond to physical phenomena that
either do or do not occur within a physical system [8], \disjoint" events (which are simple
observables) should be mutually exclusive physical phenomena.
The role of \union" will be played by sums of projections; however, just as with
intersection, E is not necessarily closed under \unions":
Proposition 3.5. If PE : H!H and PF : H!H are orthogonal projections onto closed
subspaces E and F respectively, then TFAE:
1. PE +PF is an orthogonal projection.
2. E and F are orthogonal.
3. PEPF = 0 = PFPE.
Proof. Since we have already shown that (ii)()(iii), we may use those as interchangeable.
So, we will rst assume (ii) and (iii).
To prove that PE + PF is an orthogonal projection, it will su ce to prove that it is
self-adjoint and idempotent. First, idempotent:
(PE +PF)2 = P2E +PEPF +PFPE +P2F = PE + 0 + 0 +PF = PE +PF.
3In terms of closed subspaces of H, E and F represent \disjoint" events if they are orthogonal.
54
Next, self-adjoint: Let x;y2H. Then,
h(PE +PF)x;yi=hPEx+PFx;yi=hPEx;yi+hPFx;yi
=hx;PEyi+hx;PFyi=hx;PEy +PFyi
=hx;(PE +PF)yi:
Now, assume (i), and we will show (ii). Now, by the claim in the preceeding proposition, it
will su ce to show that PF(E) =f0g. Then,
PE +PF = (PE +PF)2 = P2E +PEPF +PFPE +P2F.
So, 0 = PEPF +PFPE, and hence PEPF = PFPE. First, note that if x2E\F, then
x = PEx = PEPFx = PFPEx = Fx = x.
Hence x = 0. Now, let x2E. Then,
PEPFx = PF(PEx) = PF(x).
Since E and F are linear, ( PFx) 2E\F = f0g. Thus, PFx = f0g, and therefore, E
and F are orthogonal.
Hence, we have an analogy for \disjoint unions" (which extends readily to nite \disjoint
unions"). Intuitively, the union of two disjoint events is the event that occurs i only one of
the two events occurs. In fact, provided that E and F are orthogonal, the range of PE +PF
is E F.
Therefore, for PE;PF 2E, their \union", PE_PF is in E i E and F are orthogonal,
and hence we have
De nition 3.6. For orthogonal E;F 2E, PE_PF := PE F.
Furthermore, we can easily de ne an arbitrary union for all PE;PF 2E as PE_PF :=
PSpanE[F = PE F.
55
As for countable disjoint unions, nontrivial countable collection of pairwise disjoint
orthogonal projections will not exist in a nite dimensional vector space. However, in the
case of an in nite dimensional complex Hilbert space, we have the following proposition:
Proposition 3.7. Let H be an in nite dimensional complex Hilbert space and fPEkg a col-
lection of mutually disjoint orthogonal projections. Then,
X
k
PEk exists and is an orthogonal
projection.
Proof. Given the existence of P =
X
k
PEk, to show that it is an orthogonal projection, we
must show that it is self-adjoint and indempotent. Note rst that each partial sum
nX
k=1
PEk
is an orthogonal projection. Then, the self-adjoint and indempotent properties come from
the fact that the square function is continuous and the inner product is jointly continuous.
Therefore, E is closed under countable \disjoint unions". Henceforth, events will be
represented by both orthogonal projections and their ranges (primarily by their ranges unless
there is a risk of ambiguity); H will be denoted with 1. We will call these quantum events.
Under a partial ordering on E, where PE PF if E F for all PE;PF 2E, then E
is a lattice. In fact, given any subset A of E, inf A = PTE2AE and supA =
_
E2A
PE4.
3.2.3 A Probability
Classically, a probability p is a countably additive mapping from a -algebra of subsets
of a set into the interval [0;1] such that p( ) = 1. Hence, an analogous function de ned
on E should be a mapping p : E ! [0;1] such that for every sequence of mutually disjoint
events fEjg,
p
X
j
Ej
!
= Pjp(Ej) and p(1) = 1.
4Recall that, if the PE are not \disjoint", then the union is actually the projection onto the closed linear
span of their union
56
Instead of \probability," this mapping de ned on quantum events is often called a state
map because \[it] gives a theoretically complete description of the system. Since quantum
mechanics is a probabilistic theory, a complete description of a quantum system is given by
a probability measure on its set of events."[8] In fact, probabilities are usually determined
by states.
Example 3.8. Let P be a positive semi-de nite matrix (operator) with unit trace.5 In other
words, P is a density operator or, more speci cally, a state. For each E2E, de ne
p(E) := tr(PE).
Proposition 3.9. p : E ![0;1] satis es the following:
1. p(1) = 1
2. Given a countable collectionfEkgof pairwise disjoint events, p
X
k
Ek
!
=
X
p(Ek).
Proof. Let p be de ned as above.
1. First, p(1) = tr(P1) = tr(PI) = tr(P) = 1.
2. Finite additivity comes from the linearity of the trace map, and countable additiv-
ity comes from the fact that the trace (or Schatten) class operators6 form an ideal,
I1(H) := fT 2I1(H) :
X
n
jsn(T)j<1g, that is, the class of all operators on H
with absolutely summable singular values (sn) 7. Therefore, for any collectionfEjgof
pairwise disjoint events, tr
P
1X
j=1
Ej
!
<1.
In fact, in a separable Hilbert space of dimension at least 3, this is the only example of
a probability on E by Gleason?s theorem.
5Both properties are characterizable by eigenvalues: A positive semi-de nate matrix has all nonnegative
eigenvalues, and the trace of a matrix is the same as the sum of its eigenvalues.
6operators with nite trace
7A singular value of T is the square root of an eigenvalue of T T.
57
Theorem 3.10 ([7], Gleason). Let be a measure on the closed subspaces of a separable
Hilbert space H of dimension at least three. There exists a positive semi-de nite self-adjoint
operator P of the trace class such that for all closed subspaces E of H, (E) = tr(PPE)
where PE is the orthogonal projection of H onto E.
Restricting our positive semi-de nite self-adjoint operators of the trace class to density
operators, we get a probability measure and a special case of Gleason?s Theorem:
Theorem 3.11 ([8], Gleason). If dim(H) 3 and p is a probability on E, then there exists
a unique density operator (state) P on H such that p(E) = tr(PE) for all E2E.
3.2.4 Observables as Random Variables
To de ne integration with respect to our probability, we must rst nd an analog of a
random variable. The natural choice would be a bounded linear operator or matrix X. In
which case, we would de ne the integral as
Z
X dp = tr(XP).8
If X is Hermitian, then it is an analog to a real random variable, and the integral is
merelyEX. In the complex case, we want guarantee at least diagonalizability. Hence, normal
matrices will be our extension of complex random variables. These are our observables,
and we will denote their class by O. Note that O = span(E)9.
Since an observable is not necessarily compact, we will distinguish the space of compact
observables (i.e. with eigenvalues converging to 0) and denote the subclass by O1 := O\
I1(H), where I1(H) denotes the ideal of compact operators.
By the same token, we can also de ne moments EXk, the Fourier transform EeitX, other
transforms such as E(XY), variance Var(X) = EX2 (EX)2, covariance, etc..
8We can also denote the integral
Z
X dP since the probability is uniquely determined by the density
matrix P.
9The simplest nite-valued observable is a nite linear combination of bounded orthogonal projections.
Hence all nite-valued observables are in span(E). Therefore, spanE O. Furthermore, any observable X
can be written as the limit of a sequence of nite-valued observables, which is in span(E).
58
3.3 Quantum Measures
Now, we will begin extending our quantum \probability". First, we will extend it
to a quantum \measure". To that end, we will begin with a countably-additive function
: E !R (or C). This function can be extended to a linear mapping on span(O) in the
strong operator topology10. Hence, would be a continuous functional on I1(H) by the
Hahn-Banach Theorem.
By the Schatten Theorem11, must then be of the form
(T) = trQT , T 2I1(H)
for some unique P 2I1(H).
3.4 Quantum Vector Measures
Now, we will nd the natural extension of this notion to vector measures. To do so we
will replace the range space with a Banach space X:
F : E !X,
which, as before is extendable to F : I1(H)!X (by linearity and continuity).
By theorem 2.26, we have an immediate example of a vector measure on a measure
space (S;S; ) is given by the Bochner integral of a Bochner integrable function f : S!X:
F(E) :=
Z
E
f(s) (ds), for all E2S,
If f is simple, i.e. f := Pkxk1IEk (where xk2X and Ek2S for 1 k n),
F(E) =
X
k
xk (E\Ek) for all E2S.
We may, instead, simply use n arbitrary nite measures k :S!R (or C):
F(E) =
X
k
xk k(E) for all E2S.
10Under the assumption of continuity
11I1(H) is isometrically isomorphic to the dual of I1(H).
59
Now, to make F a \quantum vector measure", we let our \measurable space" be (H;E)
and let the k be \quantum measures": k : E !R (or C) where for each k, k(E) =
tr(QkE) for all E2E where Qk 2I1(H) is unique for each 1 k n. Hence, our simple
\quantum vector measure" is
F(E) =
X
k
xk tr(QkE) =
X
k
Z
E
xkEk d(tr(Qk )), for all E2E.12
Now, to look at a case that is not discrete, consider a Bochner integrable function f :
[0;1]!X, and the functions Q : [0;1]!I1 and TE : [0;1]!R given by TE(s) = tr(Q(s)E)
for a given E2E. We would like to de ne a \quantum vector measure" as
M(E) =
Z 1
0
f(s)TE(s) ds for E2E.
However, we must rst consider under what conditions fTE is Bochner integrable. It is
immediate that if jTE(s)j were Lebesgue integrable either kf(s)k or jTE(s)j were bounded,
then fTE would be Bochner integrable, for then
M(E) =
Z 1
0
kf(s)TE(s)kds =
Z 1
0
kf(s)kjTE(s)jds
would be the product of two Lebesgue integrable functions, one of which is bounded.
3.5 Questions
The preceeding section is merely the beginning. Of course, there is the immediate ques-
tion of su cient conditions for our quantum vector measure M being the Bochner integral
of fTE as above. Furthermore, under what conditions will Gleason?s theorem extend to the
quantum vector measure?
A next step would be in the direction of random quantum measures (in the wide sense
from Chapter 1), in which case we would choose Banach subspaces of L0( ;F;P) or C( )
(where is a \nice" topological space) for our Banach space. Furthermore, our density
12Recall that we denote our events (which are projections) by their ranges, i.e. PE is identi ed with E.
Therefore, we write Ek instead of 1IEk because, for our quantum measures, the projection itself plays the
role of an indicator function for the closed subspace Ek of H.
60
maxtirx Q would be a random matrix (operator), and, just as with quantum vector measures,
we would want to know if there is a random version of Gleason?s theorem.
61
Appendices
62
Appendix A
Measure Theory
First, we will de ne di erent types of collections of subsets: Let be a nonempty set.
De nition A.1. A ring is a collection of subsets of that is closed under nite unions and
nite intersections.
De nition A.2. A -ring is a ring that is closed under countable intersection.
De nition A.3. A -ring is a ring that is closed under countable uions.
De nition A.4. An algebra is a collection of subsets of that contains and is closed
under nite unions and relative complements.
To begin, let be a set and let be a -algebra of subsets of , that is, contains
and is closed under complements and countable unions1. Then we call ( ; ) a measurable
space, and call a subset E measurable (with respect to ) if E2 .
A.1 Measures
A measure on ( ; ) is a set function from into [0;1) for which (;) = 0 and
which is countably additive. A measure space ( ; ; ) is a measurable space together
with a measure de ned on . Note that a measure is inherently nonnegative. We call
measures that map to R \signed measures." A signed measure on ( ; ) is a set function
: ! [ 1;1] which assumes at most one of 1;1, has (;) = 0, and is countably
additive. By the Jordan Decomposition Theorem, any signed measure can be written as the
1An algebra is a collection of subsets of that contians and is closed under complements and nite
unions.
63
di erence between two mutually singular measures, + and . By mutually singular, we
mean there exist measurable sets A and B whose disjoint union is and +(A) = (B) = 0;
this is denoted + ? . Due to the existence of signed measures, nonnegative measures
will sometimes be referred to as \true" measures, but we will usually refer to true measures
as measures.
Now, let be a - nite signed measure on the measure space ( ; ; ). Then there is
a Jordan Decomposition = + on ( ; ) where +; are true measures on ( ; )
and + ? .2 Thenj j= + + is a \true" measure on ( ; ) called the variation of .
We say that measure is absolutely continuous with respect to a measure , denoted
<< , if for all E2 , if (E) = 0, then (E) = 0. Then, << i j j<< i + <<
and << .
An extremely important measure, for our purposes, is the Lebesgue measure, which we
will denote with , on (Rn;L), where L denotes the -algebra of Lebesgue measurable sets.
A measure on ( ; ) is considered nite if ( ) <1; it is considered - nite if
is the union of a countable collection of measurable sets, each of which has nite measure.
Many of our results will hold for - nite spaces. However, we will often assume that our
space is nite, or, better yet, a probability space, which we will de ne in Appendix B.
A.2 Integration and Noteworthy Theorems
A measurable function f on is called integrable if
Z
jfjd <1. Note, however,
that the integral is still de ned when
Z
jfjd 1.
The following are a few of many useful theorems from measure theory: Let ( ; ; ) be
a measure space.
2That is, the two are mutually singular, which means there exists measurable sets A and B whose disjoint
union is and +(A) = (B) = 0.
64
Theorem A.5 (Egoro ?s Theorem). Let (fn) be a sequence of measurable functions on
that converge pointwise a.e.3 on to a function f that is nite a.e. on , then for all > 0,
there exists a measurable set X such that ( X ) < and (fn) converges uniformly
to f on X .
Theorem A.6 (Fatou?s Lemma). If (fn) is a sequence of nonnegative measurable functions
on that converge pointwise a.e. to a function f on , then
Z
lim inf fn d lim inf
Z
fn d .
Theorem A.7 (Lebesgue Dominated Convergence Theorem). Let (fn) be a sequence of
measurable functions on which converge pointwise a.e. to a measurable function f on .
If there is a nonnegative integrable function g on that dominated the sequence (fn), then
f is integrable, and
limn!1
Z
fn d =
Z
f d .
There are several more, but they have their own places in this introductory chapter.
3 "a.e." or "almost everywhere" means everywhere except on a set of measure zero.
65
Appendix B
Probability Theory
B.1 Measure Theory and Probability Theory
A great deal of terms in probability theory have equivalent versions in measure theory.
So, we will begin this section pointing out these terms. First, a probability measure
space, or probability space, is a nite measure space ( ; ;P) where P( ) = 1.
The following excerpt from a table from Folland may prove useful, [6]:
Analysts? Term Probabilists? Term
-algebra - eld
Measurable Set Event
Measurable function f into (R;B) Random variable (r.v.) X
Measurable function f into (Rd;Bd);d> 1 Random vector X
Integral of f Expectation of XZ
f d E(X)
Almost every(where) (a.e.) Almost sure(ly) (a.s.)
Characteristic function Indicator function 1I
In this paper, there is a slight mixture of terms from each eld. In this chapter and
chapter 2, the language will be primarily in measure theoretic terms, with the primary
exception being the use of 1I in place of . This is to prevent confusion between the measure
theory characteristic function and the probability theory characteristic function, two very
di ernt functions1.
1And because \indicator" seems to be a more appropriate name for the function.
66
B.2 Other Basics
The following are some other useful de nitions from Durett, [4]:
De nition B.1. Let X be a random variable, then the probability measure on R (with the
Borel -algebra) induced by X is called the distribution of X, and is given by
(A) = P(X2A).2
De nition B.2. The distribution of a random variable is usually described by its distri-
bution function, F(x) = P(X x).
De nition B.3. Two random variables, X and Y are identically distributed (id),
X d= Y, if they induce the same distribution on R with the Borel -algebra.
De nition B.4. When the distribution function F(x) = P(X x) can be written as
F(x) =
Z x
1
f(y) dy
for some integrable function f, then we say that P has density function f.
Finally, we should mention the Rademacher functions, which are useful examples in
Chapter 1.
De nition B.5. The Rademacher functions are the functions rn : [0;1] ! R given by
rn(x) = sgn(sin(2n x) for all x2[0;1] for all n2Z+.
However nice the de nition is, the best understanding of the Rademacher functions
comes from just a glance:
r11
-1
1
r21
-1
1
r31
-1
1
2That is P(X 1(A)).
67
Appendix C
Radon-Nikod ym Theorem
One of the most in uential theorems from Measure theory and Probability theory is the
Radon-Nikod ym Theorem. As it is a primary focus for this paper, we will devote a slightly
larger section to it. We will discuss some of the applications of the RNT (as it is usually
abbreviated) in later sections, but rst, the standard statement of the theorem.
Theorem C.1. Let ( ; ; ) be a - nite measure space and a - nite measure on ( ; )
that is absolutely continuous with respect to . Then there exists a function f : ![0;1)
such that for all A2 ,
(A) =
Z
A
fd .
Furthermore, f is unique modulo variations on -null sets.
C.1 Radon Nikod ym Derivative
The non-negative function f is called the Radon Nikod ym Derivative, and is usually
denoted d d (which should be understood as the class of functions that are equal to f -
a.e.). Hence, we say the RNT guarantees us a nonnegative function f such that d = f d .
However, when we are working in a probability space, ( ; ;P), the RND is referred to as a
density function.
Proposition C.2. Let P be a probability measure (usually a distribution of a random variable
X) on (R;B). P is said to have density f i P << and P has RND dPd = f where is
the Lesbegue measure on (R;B).
68
Furthermore, f on Rk is called a probability density if f 0, f is measurable with
respect to k, and
Z
f d k = 1. And if a probability measure P on (Rk;B) is given by
P(A) =
Z
A
f d k for all A2B, then f2 dPd k is the density of P.
C.2 Other Forms
As the RNT is a signi cant theorem in numerous elds, it takes on di erent forms in
di erent subjects. Some forms will be developed later in this work, but for the sake of
illustration, the following some forms that are more immediate from the classic statement of
the theorem.
Theorem C.3 (RNT for Signed Measures). Let be a - nite measure on ( ; ) and
a - nite signed measure on ( ; ) such that << . Then there exists a -integrable
function f : !R such that (A) =
Z
A
fd for all A 2 . Furthermore, f is unique
modulo alterations on -null sets.
The function f is simply f + f .
Theorem C.4. Let ( ; ; ) be a nite measure space and a complex valued measure on
where << . Then there is a unique -integrable function f such that (A) =
Z
A
fd
for all A2 .
C.3 Some Signi cant Applications
C.3.1 Derivatives
The RNT introduces a notion of derivative of a (possibly signed) - nite measure
with respect to a measure . Hence when ( ; ; ) = (Rn;B; ), we can de ne the derivative
of a measure with respect to at the point x2Rn as
F(x) = limr!0 (Br(x)) (B
r(x))
69
provided the limit exists. If << , then for some -integrale f, d = f d . Then (Br(x)) (B
r(x))
is the average value of f on Br(x). If we assume nite, then we are guaranteed F = f a.e.
with respect to . When F = f a.e. with respect to , we get something like a generalization
of the Fundamtal Theorem of Calculus: the derivative of the inde nite integral of f (namely
) is f. [6]
C.3.2 Conditional Expectation
In the realm of Probability, the RND is not only the source of density functions, but
also a primary instance of conditional expectation.
De nition C.5. Given a probability space ( ; , P), 0 F a - eld, and X 2 a
random variable with EjXj<1, the conditional expectation, E(Xj 0) is the class of
random variables Y such that Y 2 0 and for all A2 0,
Z
A
X dP =
Z
A
Y dP.
If we let X 0 be a r.v. on the probability space ( ; , P), and de ne (A) = RAX dP
for all A2 , then is a measure by the the countable additivity over domains of integration
(which comes from the Monotone Convergence Theorem and linearity of integration), and
<< P by de nition. Then (A) =
Z
A
d
dP dP. Taking A = , we get
d
dP 0 integrable
and d dP is a version of E(Xj ).
If X is not nonnegative, then simply let (A) =
Z
A
X+ dP
Z
A
X dP for all A2
is a signed measure, for which we have de ned a RND.
70
Appendix D
Vector Spaces
De nition D.1. A vector space X is an abelian group with a eld K 1 and an associated
scalar product m : K X!X given by ( ;x) = x for which the following hold for all
; 2K and u;v2X:
1. ( + ) u = u+ u
2. (u+v) = u+ v
3. ( ) u = ( u)
4. 1 u = u where 1 is the multiplicative identity in K.
These spaces are also called linear spaces; however, we will adhere to the \vector"
terminology to save from confusion when we begin to de ne vector measures. We will
assume that the eld corresponding to X is K unless otherwise stated.
De nition D.2. Let S be a nonempty subset of a vector space X. The span of S (span(S))
is the collection of all linear combinations of vectors in S.
De nition D.3. The closed span of S, span(S), is the smallest closed linear ubspace
containing the span of S.
De nition D.4. A subset A of a vector space X is convex if, given x;y2A and 0 a 1,
ax+ (1 a)y2A.
In other words, for any pair of points in the space, the \line" between them is also
contained in the space. A classic example of a convex set is Sn Rn+1.
1For our purposes, K will be either R or C.
71
De nition D.5. If A is a subset of a linear space X, then the convex hull of A, co(A), is
the intersection of all convex sets containing A:
co(A) =f
nX
i=1
aixijxi2A;0 ai 1;
nX
i=1
ai = 1g.
De nition D.6. If A is a subset of a linear topological space X, then the closed convex
hull of A, co(A), is the intersection of all closed convex sets containing A.
D.1 Some Signi cant Spaces
The following are several signi cant types vector spaces. But before we introduce them,
we must rst introduce the concept of Cauchy and completeness of a metric.
De nition D.7. A sequence (xn) in a metric space X is Cauchy if for any > 0, there
exists an N2N such that for all n;m N, d(xn;yn) < .
De nition D.8. A metric is complete if every Cauchy Sequence converges with respect to
that metric.
Intuitively, it is best to understand a complete space as a space with no \holes" in it.
De nition D.9. An F-space is a metric topological vector space X over K such that,
1. The metric on X is translation invariant.
2. Scalar multiplication, : K X ! X, is continuous with respect to the metric on
K X and X.
3. Vector addition + : X X!X, is continuous with respect to the metric on X X
and X.
4. X is complete.
Save the completeness requirement, all normed vector spaces satisfy all of these criteria.
Now, for a normed vector space that is complete.
72
De nition D.10. A Banach space is a normed vector space that is complete with respect
to the norm metric.
Note that a Banach Space is an F-space wherejj xjj=j jjjxjjfor all x2X and 2K.
De nition D.11. A Hilbert space H is a vector space over C together with a function
h ; i: H H!C (known as an inner product for H) for which for all x;y;z2H and 2C
1. hx;xi= 0 i x = 0
2. hx;xi 0
3. hx+y;zi=hx;zi+hy;zi
4. h x;yi= hx;yi
5. hx;yi=hy;xi
6. If (xn)2H and limn;m!1hxn xm;xn xmi= 0, then there is an x2H with
lim
n!1
hxn x;xn xi= 0.
Given the inner product on a Hilbert Space, we can de ne a norm on the space as
jjxjj=p.
D.1.1 Example Spaces
Next, we will de ne several signi cant spaces which will be alluded to throughout this
text; all of these spaces are Banach spaces.
Let K be a scalar eld (usually R or C).
1. Let 1 p<1 and n2Z+. Then lpn =fx = (a1;:::;an) : ai in K for 1 i ng.
The norm on lpn is jjxjj=
nX
i=1
jaijp
!1=p
.
Sometimes it is written as lpn(K) to make the associated scalar eld explicit.
73
2. Let 1 p<1, then ?p =fx = (ai) in K:
1X
i=1
jaijp <1g.
The the norm on ?p is jjxjj=
1X
i=1
jaijp
!1=p
.
3. Let n 2 Z+. Then lpn = fx = (a1;:::;an) : ai in K for 1 i ng with norm
jjxjj= sup
1 i n
jaij.
4. ?1 =fx = (ai) in K: (ai) is bounded g. The norm on ?1 is given by jjxjj= sup
i
jaij.
5. c =fx = (ai) in K : ai!a for some a2Kg. The norm on c is given byjjxjj= sup
i
jaij.
6. c0 =fx = (ai) in K : ai!0g. The norm on c0 is given by jjxjj= sup
i
jaij.
7. cs =fx = (ai) inK : Pai 0, 0 < < 2r, and let x2 BY (0). Then
x0 + x2 BX (x0). Hence kT(x0 + x)kY r. But then,
kTxkY =k T( x)kY =j jkT( x+x0) Tx0kY kT( x+x0)kY +kTx0kY 2 r< .
Since T0 = 0, T is continuous at 0.
(5))(6).
80
Suppose T is continuous at 0, then choose > 0 such that T( BX (0)) BY1 (0). Let
x1;x2 2X with x1 6= x2. Then x1 x2kx1 x2k
X
2 2 B
X
(0). Hence, kT(
x1 x2
kx1 x2kX
2)kY 1. Then,
kTx1 Tx2kY =kT(x1 x2)kY =k2kx1 x2kX T( x1 x2kx1 x2k
X
2)kY
= 2kx1 x2kX kT( x1 x2kx1 x2k
X
2)kY 2= kx1 x2kY .
(6))(1)
Lipschitz continuity with x2 = 0 gives boundedness.
A slightly larger class of linear operators is the class of closed linear operators.
De nition D.20. Let X and Y be vector spaces over K, and let T : D(T) X !Y be
a linear operator. Then T is called a closed operator if its graph f(x;Tx) : x2D(T)g is
closed in X Y.
Proposition D.21. Let T be a linear operator from its domain, D(T) X into Y. Then,
T is closed i given (xn) in D(T) such that xn converges to x2X and Txn converges to
y2Y, then x2D(T) and Tx = y.
Notice that a continuous linear operator de ned on X is closed, but a closed linear
operator need not be continuous. (By the Closed Graph Theorem, if X and Y are Banach
spaces, then a linear operator T : X !Y is closed i it is continuous.) Usually, we will
require that T be densely de ned, i.e. that D(T) is dense in X.
D.3.1 Dual Spaces
De nition D.22. Given a vector space X over K, a linear functional is a linear operator
T : X!K.
De nition D.23. The dual of a vector space X is the collection of all bounded linear
functionals on X. The dual of X is denoted X and its elements are usually denoted x .
By the previous Proposition, since K is a eld it is a vector space over itself; therefore
X is a vector space with the operator norm. Furthemore, if K is a Banach Space, then so is
81
X . As for examples of Linear Operators, Linear Functionals, and Dual Spaces; an excellent
example of each can be found in the construction of the duals of Lp spaces.
For an example, consider the duals of the Lp spaces, where 1 p<1.
We begin by de ning a linear functional Tf : Lp( ; ) !K with respect to a function
f2Lq( ; ) (where q is the conjugate of p) by
Tf(g) =
Z
fg d for all g2Lp( ; ).
H older?s Inequality grants that Tf is a bounded linear functional (and is hence a member
of (Lp) ) and thatjjTfjj=jjfjjq. Hence, the linear operator T : Lq!(Lp) , give by f7!Tf
for all f2Lq, is an isometry. The isomorphism is due to Riez, and so the following theorem
is often attributed to Riez:[17]
Theorem D.24 (The Riesz Representation Theorem for the Dual of Lp( ; )). Let ( ; ; )
be a - nite measure space, 1 p <1, and q the conjugate of p. Then T is an isometric
isomorphism of Lq onto (Lp) (where T and Tf are de ned as above for all f2Lq).
In other words, we consider (Lp) = Lq for 1 p <1, and by the symmetry of the
conjugate relationship, we also have (Lq) = Lp (for p6= 1;1). Hence, (Lp) = Lp! We
would like for the same to hold true for p = 1 or p = 1. But, so far, all we have is that
(L1) = L1.
So, is L1 the pre-dual of L1?
It turns out that the dual of L1 is isometric and isomorphic to the normed linear space of
bounded nitely additive signed measures on that are absolutely continuous with respect
to . In fact, (a fact not easily veri ed) L1 does not have a pre-dual!
D.3.2 More on Operators
The following de nitions are from Linear Operators by Dunford and Schwartz
De nition D.25. The uniform operator topology in L(X;Y) is the metric topology of
L(X;Y) induced by the operator norm.
82
De nition D.26. The strong operator topology on L(X;Y) is de ned by the basis
B=fN(T;A; ) =fR2L(X;Y) :k(T R)xk< ;x2Ag: A X nite and > 0g.
De nition D.27. The weak operator topology on L(X;Y) is de ned by the basis
B = fN(T;A;B; ) = fR2L(X;Y) : jy (T R)xj< ;y 2B;x2Ag : A X;B
Y nite and > 0g
As far as relationships between the three go, the weak operator topology is contained
in the strong operator topology, which is contained in the uniform operator topology.
Now that we have de ned the dual of a Banach space, we can de ne the weak topology
on a Banach Space X.
De nition D.28. The weak topology of X is the inverse image topology generated by
X .2
With our topologies de ned, we may now name two particular types of operators that
will be used in the course of this paper.
De nition D.29. Let T 2L(X;Y) and S the closed unit sphere in X. T is a compact
linear operator if the strong closure of TS is compact in the norm topology of Y.
Equivalently, T 2L(X;Y) is compact i the image of any bounded set in X is relatively
compact in Y, i.e. its closure is compact in Y.
De nition D.30. Let T 2L(X;Y) and S the closed unit sphere in X. T is a weakly
compact linear operator if the weak closure of TS is weakly compact, i.e. it is compact
in the weak topology of Y.
Equivalently, T 2L(X;Y) is weakly compact i the image of any bounded set in X
is weakly sequentially compact, i.e. every sequence (yn) in B contains a subsequence (y0n)
which converges weakly to a point y2Y, i.e. for all y 2Y , limny y0n = y y.
2It is the smallest topology on X with respect to which all x in X are continuous.
83
D.4 Operators in a Hilbert Space
Some of the de nitions and propositions in this section can be generalized. However, we
present them in the form that will be relevant for this particular work. Let H be a Hilbert
space.
De nition D.31. A projection P : H!H is a linear operator onto a subset of H such
that P2 = P
Proposition D.32. A projection P : H!H maps H onto a closed linear subspace of H i
P is bounded.
De nition D.33. Two elements x and y of H are orthogonal if hx;yi= 0
De nition D.34. Given a closed linear subspace E of H, the orthogonal complement of
E, E? is the collection of all elements of H that are orthogonal to all members of E.
De nition D.35. Two subspaces E and F of H are orthogonal if E F? (and, conse-
quently, F E?).
Proposition D.36. For each closed subspace E of H, there exists a unique projection PE =
projE that maps H onto E and is de ned such that for each x 2H, PE(x) is the unique
element in H such that (x PE(x)) 2E?. Furthermore, if P is a continuous projection,
then R(P) is closed and R(P) R(P)? = H.
De nition D.37. An orthogonal projection is a projection such that N(P) and R(P)
are orthogonal.
De nition D.38. Let T : H!H be a linear operator. The adjoint T of T is the unique
operator on H such that hTx;yi=hx;T yi for all x;y2H.
De nition D.39. A linear operator T : H!H is self-adjoint if T = T .
84
De nition D.40. A bounded self-adjoint linear operator T : H!H is called Hermitian.
A bounded linear operator that commutes with its adjoint (i.e. TT = T T) is called
normal.
A Hermitian operator is characterized by having real eigenvalues, whereas normal op-
erators may have complex eigenvalues. (In nite dimensional cases, where operators are
matrices, normal matrices are guaranteed to be diagonalizable.)
Proposition D.41. A bounded projection T is orthogonal i it is self-adjoint3.
Proof. Let T : H!H be a bounded projection.
) Suppose T is orthogonal, and let x;y2H. Then there exist u;v2R(T)? such that
x = u+Tx and y = v +Ty. Then,
hTx;yi=hTx;vi+hTx;Tyi= 0 +hTx;Tyi=hTx;Tyi
and
hx;Tyi=hu;Tyi+hTx;Tyi= 0 +hTx;Tyi=hTx;Tyi
Hence, T = T
( Suppose T is self-adjoint. Then, N(T) = N(T ) = R(T)?.
De nition D.42. A positive semi-de nite matrix/operator is an operator T such that
for any nonzero x2H, x Hx is real and nonnegative.
De nition D.43. The trace of an n n matrix A = [aij] is given by tr(A) =
nX
i=1
aii. A
matrix A has unit trace if tr(A) = 1. More generally, the trace of a bounded operator T
is given by tr(T) :=
X
k
hTek;eki where fekg is an orthonormal basis of a separable hilbert
space H.
3An operator T is self-adjoint if T = T where T is the adjoint of T, i.e. the unique operator T : H!H
such that hTx;yi=hx;T yi.
85
If we consider B(H) (the collection of bounded operators onH) a ring, then some classes
of operators may be considered ideals. For example, the class of compact operators is an
ideal and is denoted I1.
The ideal of trace class operators, I1 = fT 2 I1 : tr(T)is absolutely convergentg,
a.k.a. Schatten operators, is the ideal of compact operators with nite trace.
86
Appendix E
Summability
In Chapter 1, we gave several characterizations of unconditional convergence. In this
section, we will give the proofs of those theorems.
First, we will de ne summability in an F-space:
De nition E.1. A sequence (xk)2XN is summable if for every > 0, there is a nite set
K N such that, for every nite L N that is disjoint with K,
X
n2L
xk
< .
Theorem E.2 (Orlicz). In a complete metrizable topological vector space, the following are
equivalent:
1. (xk) is summable.
2.
X
k2N
xk converges unconditionally in X.
3.
X
k2N
skxk converges for every sequence (sk)2f 1;1gN.
.
X
k2N
skxk converges for every sequence (sk)2f0;1gN.
Proof. Let (xk) be a sequence in a complete metrizable t.v.s. X.
1. ((3) ) ( )) Suppose
X
k2N
skxk converges for every sequence (sk) 2f 1;1gN, and let
(sk)2f0;1gN.
Let (s0k)2f 1;1gN such that for each k2N, s0k :=
8
><
>:
1 : sk = 0
1 : sk = 1
.
Then
X
k2N
xk and
X
k2N
s0kxk converge, and hence so does
87
X
k2N
xk +
X
k2N
s0kxk =
X
k2N
2skxk = 2
X
k2N
skxk.
Therefore,
X
k2N
skxk converges.
2. (( ) ) (3)) Suppose
X
k2N
skxk converges for every sequence (sk) 2f0;1gN, and let
(sk)2f 1;1gN.
Let (s k ) and (s+k )2f0;1gN such that for each k2N,
s k :=
8
><
>:
0 : sk = 1
1 : sk = 1
and s+k :=
8
><
>:
1 : sk = 1
0 : sk = 1
.
Then
X
k2N
s kxk and
X
k2N
s+kxk converge, and hence so does
X
k2N
s kxk +
X
k2N
s+kxk =
X
k2N
skxk.
3. ((1) ) (2)) Suppose (xk) is summable and : N!N is a permutation on N. Let
> 0 and K N nite such that for any nite subset L of N that is disjoint with K,
X
k2L
xk
< . Let N2N such that (k) =2K for all k N.
Then, for all m n N,
mX
k=n
x (k)
< .
4. ((2) ) ( )) Suppose that there exists a sequence (sk) 2f0;1gN such that
X
k2N
skxk
does not converge. Then there exists an > 0 such that for all N 2N, there exists
n;m N such that
X
k=nm
skxk
.
Let n1;m1 1 such that
m1X
k=n1
skxk
.
Let N2 = m1 + 1 and n2;m2 N2 such that
m2X
k=n2
skxk
.
Then, for allj 2, de neNj := mj 1+1 and letmj;nj Nj such that
mjX
k=nj
skxk
.
88
De ne Mj := fxk : nj k mj and sk = 1g. (Note that the Mj?s are disjoint.) Let
: N!N be a permutation that maps each Mj onto a block of consecutive numbers.
Then
X
k2N
x (k) diverges, and hence (xk) is not unconditionally convergennt.
5. (( ))(1)) Suppose there exists an > 0 such that for all nite K N there exists a
nite L N disjoint with K such that
X
k2L
xk
.
Let K0 =f1g and K00 N disjoint with K0 such that
X
k2K00
xk
.
Let K1 = K0[K00 and K01 N disjoint with K1 such that
X
k2K01
xk
.
Let Kn := K0n 1[Kn and K0n N disjoint with Kn such that
X
k2K0n
xk
.
Let K :=
[
k2N
Kn, and de ne (sk)2f0;1gN by sk :=
8
><
>:
0 : k =2K
1 : k2K
.
Then,
X
k2N
skxk diverges.
Theorem E.3. Let X be a Banach space. Then (xk)2XN is summable i
X
k2N
skxk where
(sk) is any bounded sequence of scalars.
Proof. The (() implication is immediate from the preceding theorem since any sequence
(sn) 2f0;1gN is bounded. For the other direction, let (ak) 2?1. Let > 0, K N nite
such that for all nite L N that are disjoint with K,
X
k2L
xk
< , and N2N such that
k