768 CHAPTER 28. THE NORMAL DISTRIBUTION

whenever ψ ∈S or G . Thus

µn (ψ) = FF−1µn (ψ)≡ F−1

µn (Fψ)→ F−1µ (Fψ)

≡ F−1Fµ (ψ) = µ (ψ). ■

The set G of S generalizes to ψ any bounded uniformly continuous function.

Lemma 28.4.5 If φ µn(t)→ φ µ (t) where {µn} and µ are probability measures defined

on the Borel sets of Rp, then if ψ is any bounded uniformly continuous function,

limn→∞

∫Rp

ψdµn =∫Rp

ψdµ.

Proof: Let ε > 0 be given, let ψ be a bounded function in C∞ (Rp). Now let η ∈C∞

c (Qr) where Qr ≡ [−r,r]p satisfy the additional requirement that η = 1 on Qr/2 andη (x)∈ [0,1] for all x. By Lemma 28.4.3 the set, {µn}

∞

n=1 , is tight and so if ε > 0 is given,there exists r sufficiently large such that for all n,∫

[x/∈Qr/2]|1−η | |ψ|dµn <

ε

3,

and since µ is a single measure, the following holds whenever r is large enough.∫[x/∈Qr/2]

|1−η | |ψ|dµ <ε

3.

Thus, ∣∣∣∣∫Rpψdµn−

∫Rp

ψdµ

∣∣∣∣≤ ∣∣∣∣∫Rpψdµn−

∫Rp

ψηdµn

∣∣∣∣+∣∣∣∣∫Rpψηdµn−

∫Rp

ψηdµ

∣∣∣∣+ ∣∣∣∣∫Rpψηdµ−

∫Rp

ψdµ

∣∣∣∣≤ 2ε

3+

∣∣∣∣∫Rpψηdµn−

∫Rp

ψηdµ

∣∣∣∣< ε

whenever n is large enough by Lemma 28.4.4 because ψη ∈S. This establishes the con-clusion of the lemma in the case where ψ is also infinitely differentiable. To consider thegeneral case, let ψ only be uniformly continuous and let ψk = ψ ∗φ k where φ k is a molli-fier whose support is in (−(1/k) ,(1/k))p. Then ψk converges uniformly to ψ and so thedesired conclusion follows for ψ after a routine estimate. ■

Here are some items which are of considerable interest for their own sake.

Theorem 28.4.6 Let H be a compact metric space. Then there exists a compactsubset of [0,1] ,K and a continuous function, θ which maps K onto H.

Proof: Without loss of generality, it can be assumed H is an infinite set since otherwisethe conclusion is trivial. You could pick finitely many points of [0,1] for K.

Since H is compact, it is totally bounded. Therefore, there exists a 1 net for H {hi}m1i=1 .

Letting H1i ≡ B(hi,1), it follows H1

i is also a compact metric space and so there exists a 1/2

net for each H1i ,{

hij

}mi

j=1. Then taking the intersection of B

(hi

j,12

)with H1

i to obtain sets