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2.13 Complements and Details 223

and, therefore, δ(ε):= sup{| f (x) − f (y)| : |x − y|≤ε}→0asε ↓ 0.

Deﬁne

(x) = f ∗ ρ

(x):=

−ε

f (x − y)ρ

(y) dy, (C11.8)

and note that, since f

(x) is an average over values of f within the

interval (x − ε, x + ε), | f

(x) − f (x)|≤δ(ε) for all x. Hence,

)

fdP

−

)

≤ δ(ε) for all n,

)

fdP−

)

≤ δ(ε),

)

fdP

−

fdP

)

≤

)

fdP

−

)

−

)

fdP−

fdP

)

≤ 2δ(ε) +

)

−

)

→ 2δ(ε)asn →∞.

Since ε>0 is arbitrary and δ(ε) → 0asε → 0, it follows that

fdP

→

fdP, as claimed. Next let F

, F be the distribution

functions of P

, P, respectively (n = 1, 2,...), and suppose (a) holds

and observe that (−∞, x]isaP−continuity set if and only if 0 =

P(∂(−∞, x]) ≡ P({x}). That is, x must be a continuity point of F.

Therefore, that (a) implies (c), follows from Alexandrov’s theorem. To

show that the converse is also true, suppose F

(x) → F(x) at all points

of continuity of a distribution function (d.f.) F. Consider a continuous

function f that vanishes outside [a, b] where a, b are points of continuity

of F. Partition [a, b] into a ﬁnite number of small subintervals whose

end points are all points of continuity of F, and approximate f by a

step function constant over each subinterval. Then the integral of this

step function with respect to P

converges to that with respect to P and,

therefore, one gets (a) by a triangle inequality.

Remark C11.1 It is straightforward to extend the proof of the equiv-

alence of (a), (b), and (c) to

, k ≥ 1. The equivalence of (c) and (d)

for the case

, k > 1, is similar, but one needs to use P-continuities

for approximating ﬁnite and inﬁnite rectangles. See Billingsley (1968,

pp. 17–18).

224 Markov Processes

The weak convergence of probability measures deﬁned in Section 2.11

(Deﬁnition 11.2) and discussed above is in fact a metric notion as we

now show. First we deﬁne the weak topology on the space of probability

measures.

Deﬁnition C11.1 The weak topology on the set P(S) of all probability

measures on the Borel sigmaﬁeld S of a metric space (S, d) is deﬁned

by the following system of neighborhoods:

N (µ

; f

, f

,..., f

; ε)



µ ∈ P(S):

)

dµ −

dµ

)

<ε∀i = 1,...,m



(µ

∈ P(S); m ≥ 1; f

are bounded and continuous on S,

1 ≤ i ≤ m; ε>0). (C11.9)

In the case (S, d) is a compact metric space, C(S) ≡ C

(S)is

a complete separable metric space under the “sup” norm  f 

∞

max{| f (x)| : x ∈ S}, i.e., under the distance d

∞

( f, g):=f − g

∞

≡

max{| f (x) − g(x)| : x ∈ S} (see Dieudonn´e 1960, p. 134). In this case

the weak topology is metrizable, i.e., P(S) is a metric space with the

metric

(µ, ν):=

∞



n=1

−n

)

dµ −

dν

)

, (C11.10)

where {f

: n ≥ 1} is a dense subset of { f ∈ C(S):  f 

∞

≤ 1}.

Exercise C11.1 Check that d

metrizes the weak topology. [Hint: Given

a ball B(µ

, r)inthed

metric with center µ

and radius r > 0, there

exists a neighborhood N (µ

: f

, f

,..., f

,ε) contained in it, and vice

versa.]

We now give a proof of Lemma 11.1 (see Section 2.11), namely,

(P(S), d

) is compact if (S, d) is compact metric.

Proof of Lemma 11.1. Let {P

: n ≥ 1} be a sequence of probability

measures on a compact metric space (S, d). Note that P(S) is metrized

by the metric d

(C11.10). Let n

1 j

( j ≥ 1) be an increasing sequence of

integers such that

1 j

converges to some number, say, λ( f

)as

j →∞. From the sequence {n

1 j

: j ≥ 1}select a subsequence n

2 j

( j ≥ 1)

2.13 Complements and Details 225

such that

2 j

converges to same number λ( f

)as j →∞. In this

manner one has a nested family of subsequences {P

: j ≥ 1}, i ≥ 1,

each contained in the preceding, and such that

→ λ( f

), as

j →∞, ∀k = 1, 2,...,i. For the diagonal sequence {P

: i = 1, 2,...}

one has

→ λ( f

)asi →∞(k = 1, 2,...).

Since { f

: k ≥ 1} is dense in C(S) in the sup-norm distance, λ extends to

C(S) as a continuous nonnegative linear functional such that λ(1) = 1,

where 1 is the constant function whose value is 1 on all of S. By the Riesz

representation theorem (see Royden 1968, p. 310), there exists a unique

probability measure P on S such that λ( f ) =

fdP∀ f ∈ C(S), and

lim

i→∞

fdP

fdP∀ f ∈ C(S).

For our next result we need the following lemmas. Let H = [0, 1]

the space of all sequences in [0, 1] with the product topology, referred to

as the Hilbert cube.

Lemma C11.1 Let (S, d) be a separable metric space. There exists a

map h on S into the Hilbert cube H ≡ [0, 1]

with the product topology,

such that h is a homeomorphism of S onto h(S), in the relative topology

of h(S).

Proof. Without loss of generality, assume d(x, y) ≤ 1 ∀ x, y ∈ S. Let

: k = 1, 2,...} be a dense subset of S. Deﬁne the map (on S into H ):

h(x) = (d(x, z

), d(x, z

),...,d(x, z

),...)(x ∈ S). (C11.11)

If x

→ x in S, then d(x

, z

) → d(x, z

) ∀ k, so that h(x

) → h(x)

in the (metrizable) product topology (of pointwise convergence) on

h(S). Also, h is one-to-one. For if x = y, one may ﬁnd z

such

that d(x, z

) <

d(x, y), which implies d(y, z

) ≥ d(y, x) − d(z

, x) >

d(x, y), so that d(x, z

) = d(y, z

). Finally, let

≡ (a

, a

,...) →

a = (a

, a

,...)inh(S), and let x

= h

−1

(

), x = h

−1

(

a). One then has

(d(x

, z

), d(x

, z

),...) → (d(x, z

), d(x, z

),...). Hence d(x

, z

) →

d(x, z

) ∀k, implying x

→ x, since {z

: k ≥ 1} is dense in S.

Lemma C11.2 The weak topology is deﬁned by the system of neighbor-

hoods of the form (C11.9) with f

, f

,..., f

bounded and uniformly

continuous.

226 Markov Processes

Proof. Fix P

∈ P(S), f ∈ C

(S), ε>0. We need to show that the

set {Q ∈ P(S): |

fdQ−

fdP

| <ε} contains a set of the from

(C11.9) (with µ

= P

,µ= Q), but with f

’s uniformly continuous and

bounded. Without essential loss of generality, assume 0 < f < 1. As in

the proof of Theorem C11.2 (iii) ⇒ (i) (see the relations (C11.3)), one

may choose and ﬁx a large k such that 1/k <ε/4 and consider the sets

in that proof. Next, as in the proof of (ii) ⇒ (iii) (of Theorem C11.3),

there exist uniformly continuous functions g

,0≤ g

≤ 1, such that

− P

)| <ε/4k,1≤ i ≤ k. Then on the set



)

dQ −

)

<ε/4k,1≤ i ≤ k



one has (see (C11.3))

fdQ≤



i=1

Q(F

) +

≤



i=1

dQ +



i=1

≤



i=1

) +

fdP

2ε

≤

fdP

+ ε.

Similarly, replacing f by 1 − f in the above argument, one may ﬁnd

uniformly continuous h

,0≤ h

≤ 1, such that, on the set



Q :

)

dQ −

)

<ε/4k,1≤ i ≤ k



one has

(1 − f )dQ <

(1 − f )dP

+ ε.

2.13 Complements and Details 227

Therefore,



)

fdQ−

fdP

)

<ε



⊃



)

dQ −

)

<ε/4k,

)

dQ −

)

<ε/4k,

for 1 ≤ i ≤ k



By taking intersections over m such sets, it follows that a neighbor-

hood N(P

)ofP

of the form (C11.9) (with µ

= P

and f

∈ C

(S),

1 ≤ i ≤ m) contains a neighborhood of P

deﬁned with respect to

bounded uniformly continuous functions. In particular, N (P

)isan

open set deﬁned by the latter neighborhood system. Since the latter

neighborhood system is a subset of the system (C11.9), the proof is

complete.

Our main result on metrization is the following.

Theorem C11.3 Let (S, d) be a separable metric space, then P(S) is a

separable metric (i.e., metrizable) space under the weak topology.

Proof. By Lemma C11.1, S may be replaced by its homeomorphic

image S

≡ h(S)in[0, 1]

. The product topology on [0, 1]

makes

it compact by Tychonoff’s theorem and is metrizable with the metric

b):=



∞

n=1

−n

− b

a = (a

, a

,...),

b = (b

, b

,...)). Ev-

ery uniformly continuous (bounded) f on S

(uniform continuity be-

ing in the

d-metric) has a unique extension

f to

(≡ closure of S

[0, 1]

f (

a):= lim

k→∞

f (

), where

∈ S

→

a. Conversely,

the restriction of every g ∈ C(

) is a uniformly continuous bounded

function of S

. In other words, the set of all uniformly continuous real-

valued bounded functions on S

, namely, UC

) may be identiﬁed

with C(

) as sets and as metric spaces under the supremum distance

∞

between functions. Since

is compact, C

(

) ≡ C(

) is a sepa-

rable metric space under the supremum distance d

∞

, and, therefore, so is

). Letting { f

: n = 1, 2,...} be a dense subset of UC

), one

now deﬁnes a metric d

on P(S

) as in (C11.10). This proves metriz-

ability of P(S

To prove separability of P(S

), for each k = 1, 2,..., let D

: i = 1,...,n

} be a ﬁnite (1/k)-net of S

(i.e., every point of S

is within a distance 1/k from some point in this net). Let D ={x

228 Markov Processes

i = 1,...,n

, k ≥ 1}=∪

∞

k=1

. Consider the set E of all probabilities

with ﬁnite support contained in D, having rational mass at each point of

support. Then E is countable and is dense in P(S

). To prove this last

assertion, ﬁx P

∈ P(S

). Consider the partition generated by the set of

open balls {x ∈ S

: d(x, x

) <

}, 1 ≤ i ≤ n

. Let P

be the probability

measure deﬁned by letting the mass of P

on each nonempty set of the

partition be assigned to a singleton {x

} in D

, which is at a distance

of at most 1/k from the set. Now construct Q

∈ E, where Q

has the

same support as P

but the point masses of Q

are rational and are such

that the sum of the absolute differences between these masses of P

and

corresponding ones of Q

is less than 1/k. Then it is simple to check

that d

, Q

) → 0ask →∞, that is,

gdQ

→

gdP

for every

uniformly continuous and bounded g on S

Our next task is to derive a number of auxiliary results concerning

convergence of probability measures, which have been made use of in

the text.

In the following lemma we deal with (signed) measures on a sigmaﬁeld

E of subsets of an abstract space . For a ﬁnite signed measure µ on this

space the variation norm µ is deﬁned by µ: = sup{|µ(A)| : A ∈ E}.

Lemma C11.3 (Scheff´e’s Theorem) Let (, E,λ) be a measure space.

Let Q

(n = 1, 2,...), Q be probability measures on (, E) that are

absolutely continuous with respect to λ and have densities (i.e., Radon–

Nikodym derivatives) q

(n = 1, 2,...), q, respectively, with respect to

λ.If {q

} converges to q almost everywhere (λ), then

lim

Q

− Q=0.

Proof. Let h

= q − q

. Clearly,

dλ = 0(n = 1, 2,...),

so that

2Q

− Q=

>0}

dλ −

≤0}

dλ

= 2

>0}

dλ = 2

· 1

>0}

dλ. (C11.12)

The last integrand in (C11.12) is nonnegative and bounded above by q.

Since it converges to zero almost everywhere, its integral converges to

zero.

2.13 Complements and Details 229

Lemma C11.4 Let S be a separable metric space, and let Q be a prob-

ability measure on S. For every positive ε there exists a countable family

: k = 1, 2,...} of pairwise disjoint Borel subsets of S such that (i)

∪{A

: k = 1, 2,...}=S, (ii) the diameter of A

is less than ε for every

k, and (iii) every A

is a Q-continuity set.

Proof. For each x in S there are uncountably many balls {B(x:δ): 0 <

δ<ε/2} (perhaps not all distinct). Since

∂ B(x:δ) ⊂{y:ρ(x, y) = δ},

the collection {∂ B(x:δ): 0 <δ<ε/2} is pairwise disjoint. But given

any pairwise disjoint collection of Borel sets, those with positive Q-

probability form a countable subcollection. Hence there exists a ball

B(x)in{B(x:δ): 0 <δ<ε/2} whose boundary has zero Q-probability.

The collection {B(x): x ∈ S} is an open cover of S and, by separability,

admits a countable subcover {B(x

): k = 1, 2,...}, say. Now deﬁne

= B(x

), A

= B(x

)\B(x

),...,A

= B(x



∪

k−1

i=1

B(x

)



,... .

(C11.13)

Clearly each A

has diameter less than ε. Since

∂ A = ∂(S\A), ∂(A ∩ B) ⊂ (∂ A) ∪ (∂ B), (C11.14)

for arbitrary subsets A, B of S, it follows that each A

is a Q-continuity

set.

To state the next lemma, we deﬁne, the oscillation ω

(A) of a complex-

valued function f on a set A by

(A) = sup{| f (x) − f (y)|: x, y ∈ A} (A ⊂ S). (C11.15)

In particular, let

(x : ε):= ω

(B(x : ε)) (x ∈ S,ε>0). (C11.16)

Deﬁnition C11.2 Let Q be a probability measure on a metric space S.

A class G of real-valued functions on S is said to be a Q-uniformity class

lim

sup

f ∈G

)

fdQ

−

fdQ

)

= 0 (C11.17)

230 Markov Processes

for every sequence Q

(n ≥ 1) of probability measures converging

weakly to Q.

Theorem C11.4 Let S be separable metric space, and Q a probability

measure on it. A family

F of real-valued, bounded, Borel measurable

functions on S is a Q-uniformity class if and only if

(i) sup

f ∈F

(S) < ∞, (C11.18)

(ii) lim

ε↓0

sup

f ∈F

{Q({ω

(x : ε) >δ})}=0 for every positive δ.

Proof. We will only prove the sufﬁciency part, as the necessity part is

not needed for the development in Chapter 2.

Sufﬁciency: Assume that (C11.18) holds. Let c , ε be two positive num-

bers. Let {A

: k = 1, 2,...} be a partition of S by Borel-measurable

Q-continuity sets each of diameter less than ε. Lemma C11.4 makes

such a choice possible. Deﬁne the class of functions

c,ε





: |c

|  c for all k



. (C11.19)

Then

c,ε

is a Q-uniformity class. To prove this, suppose that {Q

}

is a sequence of probability measures converging weakly to Q. Deﬁne

functions q

(n = 1, 2,...), q on the set of all positive integers by

(k) = Q

), q(k) = Q( A

), (k = 1, 2,...). (C11.20)

The functions q

, q are densities (i.e., Radon–Nikodym derivatives) of

probability measures on the set of all positive integers (endowed with

the sigmaﬁeld of all subsets), with respect to the counting measure

that assigns mass 1 to each singleton. Since each A

is a Q-continuity

set, q

(k) converges to q(k) for every k. Hence, by Scheff´e’s theorem

(Lemma C11.3),

lim



(k) −q(k)|=0. (C11.21)

2.13 Complements and Details 231

Therefore

sup



)

fdQ

−

fdQ

)

: f ∈

c,ε



= sup



)



(k) −



q(k)

)

|  c for all k



 c



(k) −q(k)|→0asn →∞. (C11.22)

Thus we have shown that

c,ε

is a Q-uniformity class. We now assume,

without loss of generality, that every function f in

F is centered; that is,

inf

x∈S

f (x) =−sup

x∈S

f (x), (C11.23)

by subtracting from each f the midpoint c

of its range. This is permis-

sible because

sup

f ∈F

)

fd(Q

− Q)

)

= sup

f ∈F

)

( f − c

) d(Q

− Q)

)

(C11.24)

whatever the probability measure Q

. For each f ∈ F deﬁne

(x) = inf

x∈A

f (x) for x ∈ A

(k = 1, 2,...),

(x) = sup

x∈A

f (x) for x ∈ A

(k = 1, 2,...).

(C11.25)

Note that g

, h

∈ F

c,ε

, where c = sup {ω

(s): f ∈ F}, and



f h

, h

(x) − g

(x) = ω

), for x ∈ A

(k = 1, 2,...).

(C11.26)

It follows that

d(Q

− Q) −

− g

) dQ =

−



fdQ

−

fdQ

−

d(Q

− Q) +

− g

) dQ, (C11.27)

232 Markov Processes

so that

sup

f ∈F

)

fd(Q

− Q)

)



sup



∈F

c,ε

)



d(Q

−Q)

)

+ sup

f ∈F

−g

) dQ.

(C11.28)

Since

c,ε

is a Q-uniformity class,

lim

sup

f ∈F

)

fd(Q

− Q)

)

 sup

f ∈F

− g

) dQ

= sup

f ∈F



)Q(A

)

 sup

f ∈F





)>δ

Q(A

) + δ



 c sup

f ∈F

Q({x: ω

(x : ε) >δ}) + δ,

(C11.29)

since the diameter of each A

is less than ε. First let ε ↓ 0 and then let

δ ↓ 0. This proves sufﬁciency.

Proposition C11.1 Let S be a separable metric space. A class F of

bounded functions is a Q-uniformity class for every probability mea-

sure Q on S if and only if (i) sup{ω

(S): f ∈ F} < ∞, and (ii) F is

equicontinuous at every point of S, that is,

lim

ε↓0

sup

f ∈F

(x : ε) = 0 for all x ∈ S. (C11.30)

Proof. We assume, without loss of generality, that the functions in

are real-valued. Suppose that (i) and (ii) hold. Let Q be a probability

measure on S. Whatever the positive numbers δ and ε are,

sup

f ∈F

Q({x: ω

(x : ε) >δ})  Q



x: sup

f ∈F

(x : ε) >δ



. (C11.31)

For every positive δ the right-hand side goes to zero as ε ↓ 0. Therefore,

by Theorem C11.4,

F is a Q-uniformity class.

An interesting metrization of P(S) is provided by the next corol-

lary. For every pair of positive numbers a, b, deﬁne a class L(a, b)of