convergence of probability measures by patrick billingsley 1968

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Convergence of
Probability Measures
Patrick Billingsley
Departments of Statistics and Mathematics
The University of Chicago
JOHN WILEY & SONS, New York
•
Chichester
•
Brisbane
•
Toronto
Copyright ©
1968
by John Wiley & Sons, Inc
.
All rights reserved
.
Reproduction or translation of any part of this work beyond
that permitted by Sections 107 or 108 of the 1976 United States
Copyright Act without the permission of the copyright owner
is unlawful
. Requests for permission or further information
.
should be addressed to the Permissions Department, John
Wiley & Sons, Inc
.
Library of Congress Catalog Card Number
:
68-23922
S BN

471 07242 7
Printed in the United States of America
20 19 18 17 16 15 14 13
TO MY MOTHER
Preface
Asymptotic distribution theorems in probability and statistics have from the
beginning depended on the classical theory of weak convergence of distribu-
tion functions in Euclidean space-convergence, that is, at continuity points
of the limit function
. The past several decades have seen the creation and
extensive application of a more inclusive theory of weak convergence of
probability measures on metric spaces . There are many asymptotic results
that can be formulated within the classical theory but require for their proofs
this more general theory, which thus does not merely study itself
. This book
is about weak-convergence methods in metric spaces, with applications
sufficient to show their power and utility
.
The Introduction motivates the definitions and indicates how the theory
will yield solutions to problems arising outside it . Chapter 1 sets out the basic
general theorems, which are then specialized in Chapter 2 to the space of
continuous functions on the unit interval and in Chapter 3 to the space of
functions with discontinuities of the first kind
. The results of the first three
chapters are used in Chapter 4 to derive a variety of limit theorems for
dependent sequences of random variables
.
Although standard measure-theoretic probability and metric-space topol-
ogy are assumed, no general (nonmetric) topology is used, and the few results
required from functional analysis are proved in the text or in an appendix

.
Mastering the impulse to hoard the examples and applications till the last,
thereby obliging the reader to persevere to the end, I have instead spread
them evenly through the book to illustrate the theory as it emerges in stages
.
Chicago, March 1968-
~
Patrick
Billingsley
vii
Acknowledgements
My thanks go to Soren Johansen, Samuel Karlin, David Kendall, Ronald
Pyke, and Flemming Topsoe, who read large parts of the manuscript
; the
book owes much to their detailed suggestions, and I am very grateful
. I
should also like to thank Mary Woolridge for her typing, cheerful, swift,
and error-free
.
The writing of this book was supported in part by the Statistics Branch,
Office of Naval Research, and in part by Research Grant No
. 8026 from the
Division of Mathematical, Physical, and Engineering Sciences of the National
Science Foundation
.
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CHAPTER 1
Weak Convergence in Metric Spaces
1
.
MEASURES
IN METRIC
SPACES
Let S be a metric space
. We sha study probabiity measures on the cass °
of Bore sets in S
. Here
?
is the a-fied generated by the open sets-the
smaest a-fied containing a the open sets-and a probabiity measure on
9 is a nonnegative, countaby additive set function
P
with
P(S) = 1
.
If such probabiity measures
P,,
and
P
satisfy

f
s
f dP,, *
f
s
f dP
for
every bounded, continuous rea function
f
on S, we say that
P
n
converges
weaky
to
P
and write
P,, = P
.
Our aim in this chapter is to study this
concept in detai
; we begin with some properties of individua probabiity
measures on (S, O)
.
Athough we must sometimes assume separabiity or competeness, most
of the theorems in this chapter hod for an arbitrary metric space S
. The
spaces in our appications are usuay separabe and compete
; since they
rarey have further reguarity properties, such as oca compactness, we

never impose further restrictions
.
Measures and Integras
THEOREM 1
.1
Every probabiity measure on (S,
s")
is reguar
; that is,
if A
E
9
and 8
> 0, then there exist a cosed set F and an open set G such that
F c A c G and P(G
-
F) < e
.
Proof
.
Denote the metric on
S
by
-
p(x, y)
and the distance from
x
to
A
by

p(x, A)
.t If A
is cosed, then we may take
F =
A and G =
{x
: p(x, A) < 6}
t For terminoogy and some theorems about metric spaces, see Appendix I
.
7
8
Weak Convergence in Metric Spaces
for some
b,
since the atter sets decrease to
A
as
6 1
0 . Hence we need ony
show that the cass 9 of Bore sets with the asserted property is a Q-fied
.t
Given sets
A
n
in 9, choose cosed sets
F,,
and open sets
G
n
such that

F,, C
A
n
c
G
n
and
P(G
n
-
F
n
)
< e/2n+
1
.
If
G =
U,Gn,
and if
F
-
Un<noFn,
with n
o
so chosen that
P(UnFn
-
F)
<

E/2,
then
F (
U,An
(
G
and
P(G
-
F)
<
E
.
Thus 9 is cosed under the formation of countabe
unions
; since 9 is obviousy cosed under compementation, the proof is
compete
.
Theorem 1
.1 impies that
P
is determined by the vaues of
P(F)
for cosed
sets
F
.
Theorem 1
.3 shows that
P

is determined by the vaues of
f f
dP$
for bounded, continuous rea functions
f
defined on S
. Denote by
C(S)
the
cass of such functions
f
.
It is shown on p
. 222§ that each
f
in
C(S)
is
measurabe Y
. Everything depends on the foowing resut, which shows
how to approximate the indicator (or characteristic function) I
F
of a cosed
set F by eements of C(S)
.
THEOREM 1
.2
If
F is cosed and
e

positive, there is a function f in C(S)
such that f(x) = 1 if x
E
F, f(x) = 0 if p(x, F) >
e,
and 0 < f(x) < 1 for a
x
. The function f may be taken to be uniformy continuous
.
'
Proof
.
Define a continuous function
99
of a rea variabe by
1

if t<0,
-t

if
0<t<1,
~O'

if 1 <
t
.
If
9'(t)
=

(1
.2)

AX)
= 9'
(1
p(x, F')~
e
t
We have defined the cass
9'
of Bore sets as the a-fied generated by the open sets, which
is the same thing as the a-fied generated by the cosed sets and is the one appropriate for
the present theory
. For reated (mosty inappropriate) a-fieds, see Probem 6
.
$ When it is the entire space, we omit the region of integration
.
§ Of Appendix II, a misceany to which most measurabiity questions are reegated
.
Measures in Metric Spaces
9
then
f
has the required properties-it is even uniformy continuous
. The
drawing graphs this
f
for
F

[a, b]
on the ine
.
THEOREM 1
.3
Probabiity measures P and Q
-
on (S, 9) coincide if
(1
.3)

JfdP__JfdQ
for each f in
C(S)
.
Proof
.
Suppose
F
is cosed
. Start with
(1 .1)
and define, for each positive
integer u,
(1
.4)

9qu(t) = p(ut)
and
(1

.5)

fu(x) = pu(p(x,
F))
.
Then
{f
u
}
is a nonincreasing sequence of eements of C(S) converging point-
wise to
I
F
.
By the bounded convergence theorem,
P(F) =
im
u
f f
u
dP
and
Q(F) =
im
u
f f
u
dQ,
so that, if
(1

.3)
hods for a
f
in
C(S),
P(F) =
Q(F)
.
Since
P
and
Q
agree for a cosed sets, it foows by Theorem 1
.1 that
P
and
Q are identica
.
Thus the vaues of
f f
dP
for
f
in C(S) competey determine the vaues of
P(A)
for
A
in Y
. This fact underies the circe of ideas centering on the notion
of weak convergence

; athough we have defined weak convergence by requir-
ing the convergence of the integras of functions in C(S),
in the next section
we sha characterize it in terms of the convergence of the measures of certain
sets
.
Tightness
The foowing notion of tightness proves important both in the theory of weak
convergence and in its appications
. A probabiity measure
P
on
(S, 9)
is
tight
if for each positive s there exists a compact (p
. 217) set K such that
P(K) > 1
-
e
. Ceary, P is tight if and ony if it has a a-compact support
.]'
By Theorem 1
.1,
P
is tight if and ony if
P(A)
is, for each
A
in p, the

supremum of
P(K)
over the compact subsets K of A
.
In a space that is a-compact, every probabiity measure is tight-which
covers k-dimensiona Eucidean space
. The foowing resut, which aso
covers the Eucidean case, is more usefu
.
t A support of a probabiity measure is a set
A
in
So
with
P(A) =
1
; a set is a-compact
if it can be represented as a countabe union of compact sets
. The characterization of a
tight P as having a a-compact support is inappropriate as a definition because it does not
generaize in the right way to famiies of probabiity measures (see Section 6)
.
10
Weak Convergence in Metric Spaces
THEOREM
1
.4
If S
is separabe and compete, then each probabiity
measure on (S, p

')
is tight
.
Proof
.
Since S is separabe, there is, for each
n,
a sequence A,
n1
, A
nt
, . . .
of
open 1/n-spheres covering
S
.
Choose
i
n
so
that P(
U
i<i
n
Ani) > 1
-
e/2n
. By
the competeness hypothesis, the totay bounded set
I

In>1Ui<i,nAni
has
compact cosure
K
(see p
. 217)
. Since ceary
P(K) > 1
-
E,
the theorem
foows
.
Theorem 1
.4 is fase without the hypothesis
of
competeness
; whether the
hypothesis of separabiity can be suppressed is equivaent to the probem of
measure
. These matters are discussed in Appendix III
.j'
Remarks
.
Theorem 1
.4 is due to Uam (see Oxtoby and Uam (1939))
; LeCam (1957)
introduced the term "tight
."
PROBLEMS

.
;
1
.
Say that a function
f
separates sets
A
and
B
if
f (x) =
0 for
x
in
A,
f (x) = 1 for
x
in
B,
and 0 < f (x) <
1 for a
x
.
If
A
and
B
are at positive distance, they can be separated
by a uniformy continuous

f
[Theorem
1
.2]
.
If
A
and
B
have disjoint cosures but are
at distance 0, they can be separated by a continuous
f [f(x)
= p(x,
A)/(p(x, A) +
p(x,
B))]
but not by a uniformy continuous
f
.
There is no continuous
f
separating
A
and
B
if their
cosures meet
; there is no
f
separating

A
and
B
if they meet
.
2
. Give exampes of distinct topoogies that give rise to the same cass of Bore sets
.
3 . If S can be embedded as an open set in some compete metric space, then [Keey
(1955, p . 207)] it is topoogicay compete
. Since a ocay compact S is open in its
competion, it is topoogicay compete
. Hence Theorem 1
.4 appies if S is separabe and
ocay compact . Since such an S is a-compact [being a union of open sets with compact
cosures and hence (p
. 216) a countabe such union], it aso foows directy that each proba-
biity measure on it is tight
; Eucidean space is an exampe
.
4
. Let S be a Hibert space with a countaby infinite orthonorma basis
x
1
, x
2
, . .
. .
Since S is separabe and compete, Theorem 1
.4 appies

. However,
-
no set with nonempty
interior is compact [a nonempty interior must, for some x and e, contain a the points
x +
6x],
so that
S
is neither ocay compact nor [Baire's category theorem
; Keey (1955,
p
. 200)] a-compact
. If P assigns positive mass to each eement of a countabe, dense set,
then P has no support ocay compact in the reative topoogy
.
5
. Adapt Probem 4 to the genera Banach space of countaby infinite dimension [there
exist points
x
1
,
x
2
, . . .
with sup
ra
~Ix
n
JJ
< oo and infra#n

II
x
m
-
xn
> 0
;
see Banach
(1932, p
. 83)]
;
C[0, 1], important in probabiity, is such a space, which expains why a
theory based on oca compactness is of sma utiity in this subject . (See aso Probem 5 in
Section 3
.)
t Athough Theorem 1
.4 as given suffices for a the appications in this book, it is natura
to inquire after extensions
. It is to questions of just this sort that Appendix III is devoted
.
I
Some probems invove concepts not required for an understanding of the text itsef
;
there are no probems whose soutions are used ater in the text
. A simpe assertion is
understood to be prefaced by "show that
." Square brackets contain hints or indications of
soutions
.
Properties of Weak Convergence

11
6
. We have defined <5
0
as the a-fied generated by the open sets, which we can indicate by
writing .9' = a(open sets)
. In the same way, define
.50
1
= a(cosed
G
5
sets) (a set is
a G
a
if it is a countabe intersection of open sets), define ° °
2
=
a(C(S))
(the smaest a-fied with
respect to which each function in
C(S)
is measurabe), and define
.9'
3
= cr(open spheres),
b4
= a(compact sets), and
`9'5
= a(compact G

b
sets)
. In a metric space each cosed set is
a G
5
.
Use this fact and Theorem 1
.2 to prove
.5_Y1=
°
2
D
53
:D
Y
4
=5
0
5
.
Show that 9 =
93
if
S
is separabe
. Show that
So
= 50
5
if

S
is a-compact (which wi be
true if
S
is separabe and ocay compact)
. We may have °
"2
5`
.9
3
(even if
S
is ocay
compact)
:
Take
S
uncountabe and discrete
. We may have
Y
3
54
.5
4
(even if
S
is separabe
and compete)
: Take S to be the Hibert space in Probem 4
. (The situation differs in the

genera topoogica space, where one must consider two casses of sets
: The Bore sets are
taken as the eements sometimes of
50
and sometimes of 5°
4
,
and the Baire sets are taken as
the eements sometimes of
.S°
2
and sometimes of Y
5
the terminoogy varies
.)
7
. In connection with tightness, this fact is interesting
: Suppose
P
is defined on
(S,
.50),
but suppose at the outset ony that it is finitey additive
. If, for each
A
in
.50, P(A)
= sup
P(K)
with

K
ranging over the compact subsets of
A,
then
P
is countaby additive after a
.
2
.
PROPERTIES OF WEAK CONVERGENCE
We have defined
P
n
=> P
to mean that
f f
dP
n

f f
dP
for each
f
in the cass
C(S) of bounded, continuous rea functions on S
. Note that, since the
integras
f f
dP
competey determine

P
(Theorem 1
.3),
the sequence
{P
n
}
cannot converge weaky to two different imits at the same time
. Note aso
that weak convergence depends ony on the topoogy of S, not on the specific
metric that generates it
:
Two metrics generating the same topoogy give rise
to the same casses
Y
and
C(S)
and hence to the same notion of weak
convergence
.'
Portmanteau Theorem
The foowing theorem provides usefu conditions equivaent to weak
convergence
; any one of these conditions coud serve as the definition
. A set
A
in
Y
whose boundary
aA

satisfies
P(aA)
= 0 is caed
a
P-continuity set
(note that
aA
is cosed and hence ies in 5)
.
THEOREM 2
.1
Let P
n
,
P be probabiity measures on (S,
.)
.
These five
conditions are equivaent
:
t If we topoogize the space Z(S) of a probabiity measures on (S,
.90) by taking as the
genera basic neighborhood of
P
the set of
Q
such that
I f f
Z
dP

-
$
f
z
dQI <
E
for i =
1,
. .
. ,
k,
where
E
is positive and the
f
Z
ie in
C(S),
then weak convergence is convergence
in this topoogy
. The topoogica structure of
Z(S),
which wi be of no direct concern to
us, is discussed in Appendix III
.
12
Weak Convergence in Metric Spaces
(i) P
n
P

.
(ii) im
n
f f
dP
n
= f f
dP
for a bounded, uniformy continuous rea
f
.
(iii)
im sup
ra
P
n
(F) < P(F) for a cosed F
.
(iv) im inf
ra
P
n
(G) > P(G) for a open G
.
(v) im
n
P
n
(A) = P(A) for a P-continuity sets A
.

A coupe of exampes wi show the significance of these conditions
. Let
P
be a unit mass at the point
x (P(A)
is
1 or 0, according as
x
ies in A or
note'), and et
P
n
be a unit mass at
x
n
.
If
x
n
>
x,
then
f f
dP
n
=
f(x
n
)
-*

f(x) =
f f
dP
for a
f
in
C(S),
so that
P,, => P
.
If
x
n
does not converge to
x,
then,
for some positive e, we have
p(x
n
,
x) >
e for infinitey many
n
.
If
f(y)
qJ(s p(x, y))
with
99
defined by

(1
.1),
then
f
c-
C(S)j(x)
=
1, and
f(x,,)
= 0
for infinitey many
n
;
hence
P
n
cannot converge weaky to
P
.
Thus
P
n
=>
P
if and ony if
x,,,
-> x,
which provides an exampe we sha often use
. (Many
putative weak-convergence theorems that are in fact not theorems can be

disproved by speciaizing this exampe
.) Since A is a P-continuity set if and
ony if
x
0
aA,
it is easy to check the equivaence of (i) and (v) in this case
.
If
x
n
-+
x
but the
x
n a differ from
x,
then there is strict inequaity in (iii) for
F
{x}
and strict inequaity in (iv) for the compementary set
G = Fc
;
moreover, if the
x
n
are a distinct and
A = {x2,
X4
. . . .

},
then
P,(A)
does
not converge to
P(A)
or to anything ese
.
On the ine with the ordinary metric, the DeMoivre-Lapace theorem aso
iustrates the conditions in the theorem
. For a simper exampe equay
reevant, consider the measure
P
n
corresponding to a mass of
1/n
at each of
the points
i/n, i = 1,2,
.
.
. ,
n
. Now P
n
converges weaky to Lebesgue
measure
P
confined to the unit interva, as foows from the fact that
f f

dP
n
is an approximating sum to
f f
dP
viewed as a Riemann integra
. If
A
consists
of the rationas, then
P,(A)
=
1 does not converge to
P(A)
7-

0
;
if
G is
an
open set containing the rationas and having Legesgue measure near 0,
then there is strict inequaity in
(iv)
.
We prove Theorem
2
.1
by estabishing the impications in the foowing
diagram

.
I
(i) -> (ii) >
(iii) H
(iv)
I
(v)
Of course, (i)

(ii) is trivia
.
Proof of (ii)
>
(iii)
.
Suppose (ii) hods and that
F is
cosed
. Suppose
6 > 0
. For sma enough e,
G = {x
: p(x, F) <
e}
satisfies
P(G) < P(F) + S,
t Each subset of S mentioned is assumed to
ie
in
Y .

Properties of Weak Convergence 13
since the sets of this form decrease to F as s ,[
. 0
. If f(x) is the function
defined by (1
.2), then f is uniformy continuous on S, f(x) = 1 on F, f(x) - 0
on the compement Ge of G, and 0 < f(x) < 1 for a x
. Since (ii) hods, we
have im,, f f dPn = f f dP, which, together with the reations
P,, (F) = JF f dPn < J f f dPn
and
J f dP =J f dP < P(G) < P(F) + 6,
G
impies
im sup,, P,,(F) < imn J f dPn =J f dP < P(F) + b
.
Since 6 was arbitrary, (iii) foows
.
Proof of (iii) -* (i)
. Suppose that (iii) hods and that f c- C(S)
. We sha
first show that
(2
.1)

im supra f f dPn < f f dP
.
By transforming f ineary (with Ja positive coefficient for the first-degree
term), we may reduce the probem to the case in which U < f(x) < 1 for a
x

. For an integer k, temporariy fixed, et Fi be the cosed set Fi
{x
: ik < f(x)}, i = 0, 1,
.
.
. , k
. Since 0 < f(x) < 1, we have
k - 1

t- 1



k i

i- 1
i=1 k P{x
:-

k <f(x) < k} < f fdP <~ kPCx
:

k <f(x) < k}
.
The sum on the right is
Z~ k [P(Fi-1) - P(Fi)] = k + k i P(Fi)
.
71
This and a simiar transformation of the sum on the eft yied
(2

.2)

1 1 P(Fi) < f dP < 1 + 1 P(Fi)
.
k i=1

f

k k i=1
If (iii) hods, then im sup,, P,,(FF) < P(FF) for each i and hence (appy the
right-hand inequaity in (2 .2) to P,, and the eft-hand one to P)
('
im SUP
. f dPn I < + rf dP
.
Letting k - co, we obtain (2
.1)
.
Appying (2
.1) to -f yieds im infra f f dPn > f f dP, which, together
with (2
.1) itsef, proves weak convergence
.
The equivaence of (iii) and (iv) foows easiy by compementation
.
14
Weak Convergence in Metric Spaces
Proof of (iii)
k
(v)

.
Let
A
°
denote the interior of
A,
and et
A
-
denote its
cosure
. If (iii) hods, then so does
(iv),
and hence, for each A,
(2
.3)

P(A
-
)
> im
sup,,,
P,
,
(A
-
)
> im sup,,
P,,(A)
> im inf

z
P,,,(A)
> im inf,
P
n
(A°) > P(A°)
.
If
P(aA) = 0,
then the extreme terms equa
P(A)
and im
n
P
J
A) = P(A)
foows
.
Proof of (v)
>-
(iii)
.
Since
a{x
:
p(x, F)
<_ 8}
is containedt in
{x
: p(x,

F) =
8},
these boundaries are disjoint for distinct
a,
and hence at most countaby
many of them can have positive P-measure
. Therefore, for some sequence
of positive 6
7
,
going to 0, the sets
F
k
= {x
: p(x, F) < ok}
are P-continuity sets
.
If (v) hods, then im sup
ra
P,,(F)
<
im,,
P
f
(F
k
) = P(F
k )
for each k
;

if
F
is cosed, then
F
k
J, F,
so that (iii) foows
. This competes the proof of
Theorem 2
.1
.
Other Criteria
It is sometimes convenient to prove weak convergence by showing that
P,
JA) * P(A)
for some specia cass of sets
A
.
THEOREM 2
.2
Let GI be a subcass of
9P
such that (i) °' is cosed under the
formation of
finite intersections and (ii) each open set in S is a finite or countabe
union of eements of
QI
.
If P,
a

(A)
>
P(A)
for every
A
in
G',
then P,
a
=
:>
P
.
Proof
.
If A
1
,
. .
. , A,,, ie in
°,
then so do their intersections
; hence, by
the incusion-excusion formua,
m
Pn
U
A)
= F'iP,(Ai)
-

Y_
i7
P
n(Ai
A
j) +
Y-iikPn(AiA
;Ak)
-
.
.
i=1
IiP(Ai)
-
Ei,P(AiA,)
+ EiykP(AiA,Ak)
= P
U
A
i
0

.
=1
If G is open, then G
= U
iAi for some sequence
{Ai of eements of
1&
.

Given
s, choose
m
so that
P(Ui < mAi) > P(G)
-
E
.
By the reation just proved,
P(G)
-
e <
P(U
i
<<,nAi)
-
im
n
P
n
(U
i
<
mAi)
< im inf,
n
P,JG)
.
Since
8

was
arbitrary, condition (iv) of the preceding theorem hods
.
Let
S(x, e)
denote the (open) e-sphere about
x
.
COROLLARY 1
Let °( be a cass of sets such that (i) W is cosed under the
formation of finite intersections and (ii) for every x in S and every positive e
t The incusion may be strict-in a discrete space, for exampe
.
Properties of Weak Convergence 15
there is an
A
in
(
with x
E
A
°
~= A ~ S(x, e)
. If S is separabe and if
P,"(A)
P(A)
for every
A
in i, then
P,,, = P

.
Proof
.
Condition (ii)t impies that, for each point
x
of an open set G,
x
E
A
°
c A c
G for some
A
in
V
.
Since
S
is separabe, there exists (see p
.
216) in
V
a finite or infinite sequence
{A
i
}
such that G
(=
UiA°
and

A
i
(
G,
which impies G
=
U
i
A
i
.
Thus
G
satisfies the hypotheses of Theorem 2
.2
.
COROLLARY 2
Suppose that, for each finite intersection A of open spheres,
we have P,
JA)
*
P(A),
provided A is a P-continuity set
. If S is separabe,
then P
n
=>
P
.
Proof

.
The boundaries
aS(x,
e),
being contained in the sets
{y
:
p(x, y) = e},
are disjoint (for fixed x) and hence have P-measure 0, with at most countaby
many exceptions
. Since
a(A
r)
B) (aA)
v (aB),
it foows that the hypotheses of Coroary 1 are satisfied by the cass
G&
of
those P-continuity sets that are finite intersections of spheres, and the resut
foows
.
Let us agree to ca a subcass Y
,
'
of ° a
convergence-determining cass
if
convergence
P
n

(A)
-> P(A)
for a P-continuity sets
A
in YV' invariaby entais
the weak convergence of
P
n
to
P
.
Coroary 2 becomes
: In a separabe space,
the finite intersections of spheres constitute a convergence-determining cass
.
Let us further agree to ca
'Y' a
determining cass
if
P
and
Q
are identica
whenever they agree on 'V
. The cass of cosed sets is a determining cass and
so is any fied that generates Y
. Athough each convergence-determining
cass is ceary aso a determining cass, the foowing exampe shows that
the converse fais
. Let S be the haf-open interva [0, 1) with the ordinary

metric
; et
''-
be the cass of sets
[a, b)
with 0 <
a < b <
1
. Then 'K is a
determining cass but not (as may be seen by taking
P,, [P]
a unit mass at
1 - 1 /n [0])
a convergence-determining cass
. Athough this one is artificia,
we sha see that the appications abound with rea exampes of determining
casses that are not convergence-determining casses
.
We cose this section with another condition for weak convergence
. A
sequence {x,
z
}
of rea numbers converges to a imit x if and ony if each
subsequence
{x
.
.}
contains a further subsequence {x,
z

.}
that converges to
x
.
(It is convenient to denote a sequence of integers by
{n'}
rather than
{n k }
and a subsequence of
{n'}
by
{n"}
rather than
{n
k
,} .)
From this fact it is easy
to deduce a weak-convergence anaogue
.
f
This condition is sighty stronger than the requirement that the interiors of the eements
of
I&
form a base for the topoogy of S
.

convergence of probability measures by patrick billingsley 1968

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