Probability measures on metric spaces

282 Pages·1967·10.697 MB·English

by  Parthasarathy, K. R

Probability and Mathematical Statistics
A Series of Monographs and Textbooks
Edited by
Z. W. Birnbaum E. Lukacs
University of Washington Catholic University
Seattle, Washington Washington, D.C.
1. Thomas Ferguson. Mathematical Statistics : A Decision Theoretic
Approach. 1967
2. Howard Tucker. A Graduate Course in Probability. 1967
3. K. R. Parthasarathy. Probability Measures on Metric Spaces. 1967
In preparation
B. V. Gnedenko, Yu. K. Belyayev, and A. D. Solovyev. Mathematical
Methods of Reliability Theory
PROBABILITY MEASURES
ON METRIC SPACES
K. R. Parfhasarathy
DEPARTMENT OF PROBABILITY AND STATISTICS
THE UNIVERSITY OF SHEFFIELD
SHEFFIELD, ENGLAND
1967
ACADEMIC PRESS New York and London
COPYRIGHT © 1967, BY ACADEMIC PRESS INC.
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PREFACE
Ever since Yu. V. Prohorov wrote his fundamental paper
"Convergence of random processes and limit theorems in the theory of
probability" in the year 1956 [33], the general theory of stochastic
processes has come to be regarded as the theory of probability measures
in complete separable metric spaces. The subsequent work of A. V.
Skorohod of which a detailed account is given in the book by I. I.
Gikhman and A. V. Skorohod [9] gave a great impetus to the subject of
limit theorems in probability theory. The present monograph deals with
the general theory of probability measures in abstract metric spaces,
complete separable metric groups, locally compact abelian groups,
Hubert spaces, and the spaces of continuous functions and functions with
discontinuities of the first kind only. Emphasis is given to the work
done by the Indian School of probabilists consisting of, among others,
V. S. Varadarajan, R. Ranga Rao, S. R. S. Varadhan, and the author.
In Chapter I a detailed description of the Borel σ-field of a metric
space is given. The isomorphism theorem which states that two Borel
subsets of complete separable metric spaces are isomorphic if and only
if they have the same cardinality is proved. The by now classical theorem
of Kuratowski on the measurability of the inverse of a measurable map
from a Borel set of a complete separable metric space into a separable
metric space is established.
Chapter II deals with properties such as regularity, tightness, and
perfectness of measures defined on metric spaces. The weak topology
VI PREFACE
in the space of probability measures is investigated and Prohorov's
theorems on metrizability and compactness are proved. Convergence of
sample distributions and existence of nonatomic measures in separable
metric spaces are investigated.
Chapters III and IV deal with the arithmetic of probability distribu-
tions in topological groups. The nature of idempotent measures, existence
of indecomposable and nonatomic indecomposable measures, .and
representation of infinitely divisible distributions and its relevance in the
theory of limit theorems for "sums" of infinitesimal summands are
discussed in great detail.
Chapter V contains the proofs of the classical extension theorems
and existence of conditional and regular conditional probabilities in
standard Borel spaces. This is included for the sake of completeness.
Chapter VI gives a detailed account of the subject of limit theorems
in a Hubert space. Prohorov's theorems on the description of weakly
compact subsets of the space of measures on a Hubert space [33],
Sazonov's criterion for a characteristic function [37], and Varadhan's
results on the Lévy-Khinchine representation of infinitely divisible laws
and limit theorems for sums of infinitesimal summands [45] are estab-
lished.
Chapter VII is devoted to a study of probability measures on the
spaces C[0, 1] and D[0, 1]. Properties of the Skorohod topology [9, 20]
in the space D[0, 1] are proved. The compactness criteria for sets of
probability measures and their applications to testing statistical hypo-
theses are given.
The account of the subject given herein is by no means complete.
The topic of measures on Lie groups and homogeneous spaces, for
example, is not touched on at all. Some of. the references on this subject
have been included, however, and for further comments the reader may
refer to the bibliographical notes at the end of the book.
Theorem numbers refer to theorems in the same chapter if they are
not accompanied by a chapter number.
The author wishes to express his thanks to R. Ranga Rao, V. S.
Varadarajan, and S. R. S. Varadhan with whom he had the opportunity
of collaboration over several years. It is a pleasure to offer thanks to
F. Tops0e and O. Bjornsson who read the first two chapters of the
manuscript carefully and made valuable suggestions. He is grateful to
V. Rohatgi and K. Vijayan who also helped with proofreading.
PREFACE Vil
He is deeply indebted to the Research and Training School of the
Indian Statistical Institute, Calcutta, for providing facilities to carry
out research work during 1960-1965, a substantial portion of which has
been included in the present monograph. He is grateful to the
Department of Probability and Statistics of the University of Sheffield
for the invaluable assistance rendered in the preparation of the book.
Thanks are finally due to Miss J. Bowden for her efficient typing of
the manuscript.
Sheffield, England K. R. PARTHASARATHY
March 1967
I
THE BOREL SUBSETS OF A METRIC SPACE
1· GENERAL PROPERTIES OF BOREL SETS
Here we shall concern ourselves with the study of the properties
of Borel subsets of metric spaces. The precise metric which gives rise
to the topology will be usually unimportant. Generally speaking all
we shall use is the fact that the topology arises through a metric.
Let X be any metric space. We define by &8 , or 38 when no con-
X
fusion can arise, the smallest cr-algebra of subsets of X which contains
all the open subsets of X. 38 is called the Borel σ-field of X and elements
x
of @ are called Borel sets. @t satisfies the following conditions:
x x
(1) Xea φε@ ;
X9 χ
(2) A e& implies that A' e âù where A9 stands for the comple-
x x
ment of A ;
oo oo
(3) Av A2>..., e@x implies that LM*G^A> C\Aie@x
1 1
Since every closed set is the complement of an open set and vice
versa, St is also the smallest σ-algebra of subsets of X which contains
x
all the closed subsets of X.
We shall now examine how far the metric nature of the space X
allows us to reduce the number of axioms in the definition of ät.
x
1
2 I. THE BOREL SUBSETS OF A METRIC SPACE
Definition LI
A set A C X is called a G if it can be expressed as the intersection
ô
of a denumerable sequence of open sets. A set A Ç X is called an F
a
if it can be expressed as the union of a denumerable sequence of closed
sets.
Let d be the metric in X. For any set A CX and x e X we shall
write d(x, A) = ini d(x,y). d(x, A) is called the distance of the
yeA
set A from the point x.
Theorem LI
The function d(x, A) satisfies the inequality
\d{x,A) -d(y,A)\<d(x,y).
In particular, d(x, A) is uniformly continuous.
PROOF. By the triangular inequality we have for any ze A and any
x,yeX,
d(xA)<d{x,z)
t
^d(x,y) +d(y,z).
Taking infimum over zeA we obtain
d(x,A)<d(x,y) + d(y,A).
By the symmetry of d(x,y) we have
d(y,A)^d{x,y) +d(xA).
f
The two preceding inequalities yield
\d(x,A)-d(y,A)\^d(x,y).
Theorem L2
Every closed subset of X is a G and every open set is an F .
a a
PROOF. Let A C X be closed. It is clear that d(x, A) = 0 if and only
if XEA. Thus
A= n{*:<*(*M)<l/*}.
n=l
1. GENERAL PROPERTIES OF BOREL SETS 3
The continuity of d(x, A) (Theorem 1.1) implies that the sets
{x: d(x, A) < I/n} are open, and hence A is a G . Since the complement
ô
of a G is an F , the second part of the theorem follows from the first.
ô a
This completes the proof.
Theorem 13
Let X be any metric space. Then @l is the smallest class of subsets
x
of X which contains all the open (closed) subsets of X, and which is closed
under countable unions and countable intersections.
PROOF. Let 2 be the smallest class of subsets of X containing all
the open sets and closed under countable unions and intersections.
Since âi has both the properties, we have 2 Q<%. It is enough to show
x x
that -2 is closed under complementation. Let 2' be the class of all
sets A such that A' e 2. By Theorem 1.2 every closed set is a G , and
ô
hence belongs to 2. Thus every open set belongs to 2', which is closed
under countable intersections and unions. Hence 2 C 2'. This implies
that 2 is closed under complementation.
Theorem \A
Let X be any metric space. Then 31 is the smallest class of subsets
x
of X which contains all the open subsets of X, and which is closed under
countable intersections and countable disjoint unions.
PROOF. Let Jt?0 be the smallest class containing the open sets and
closed under countable intersections and countable disjoint unions.
In order to prove the theorem it is enough to show that Jf is closed
0
under complementation.
By Theorem 1.2 it is clear that ^ contains all closed sets. Let
0
*r = {E: E is either closed or open}. L e t^ = {E: E e^ , E' eJf }.
0 0
Then ΊΤ Q3tf Qj^ . We shall now prove that 3tf is closed under
x 0 x
countable unions and countable intersections. Indeed, let Ε £34?
{ ν
i = 1, 2,... . Since£,6^0, i = 1, 2,..., ΓΙΓ-ι £.-e*V (ίΤ-ι *.·)' =
U*°°= i Ei = Uf= l (Ei n Ei n E2 n · · · n Ei:-1) is a countable disjoint
union of sets in J^ and hence belongs to Ji? . Thus ΠΓ=ι E sJtf
0 0 i v
Uf= l Ei = LK°°= l (Ei n Ei n E2 n · - · n E'i -1) is a countable disjoint
union of sets in Jf and hence belongs to J^ . ((Jf= ι EiY = Πί= ι Ei
0 0
4 I. THE BOREL SUBSETS OF A METRIC SPACE
is a countable intersection of sets in ^f and hence belongs to ^ .
0 0
Thus \jT=iEie^v Thus JP =JP This completes the proof.
0 V
The next two theorems relate the σ-algebra âi and the continuous
x
functions on X.
Theorem L5
LetX and Y be metric spaces and f a map of X into Y.Iff is continuous,
then f is measurable.
PROOF. For every open set U ÇY, /_1(£7) is open in X, and hence
belongs to@. This implies that for every Borel set B ÇY, /-1(£) e& .
x x
Hence / is measurable.
Theorem L6
// X is a metric space and A and B are two disjoint closed subsets of
X, then there exists a continuous function f(x) on X with the properties:
(1) 0</(*)<l; (2) f(x) = 0 for xeA, = 1 for XGB. If
ù^xeA.yeB d(%, y) = à > 0, then the function f can be chosen to be uniformly
continuous.
PROOF. Let f(x) = d{x, A)j[d(x, A) -f- d(x, B)]. It is easy to verify
that / satisfies (1) and (2). The second part of the theorem follows
immediately from Theorem 1.1 and the fact that d(x, A) + d(x, B) ^ δ.
Theorem L7
If X is a metric space, the a-algebra 3$ is the smallest a-algebra of
x
subsets of X with respect to which all real-valued bounded continuous
functions on X are measurable.
PROOF. Let Jf be the smallest σ-algebra with respect to which all
real valued bounded continuous functions are measurable. By Theorem
1.5, 34? Q&. We shall now prove that every closed set C belongs to
x
JJT. By Theorem 1.2, C = ΠΓ Un where ui> U ,..., are open sets
2
and U 2 U 2,... . Since C and X — U are disjoint closed sets,
1 2 n
by Theorem 1.6, there exists a continuous function f (x) such that
n
0 < f (x) < 1 and f (x) = 1 for all x s C and = 0 for all x e X - U .
H H n
Let / = 2ί° 2~7η· Since the series is uniformly convergent, / is also