1928 CHAPTER 59. BASIC PROBABILITY

and so, considered as elements of S∗,

F−1 (µ) = φ µ (·)(2π)−(p/2) ∈ L∞.

By the dominated convergence theorem

(2π)p/2 F−1 (µn)(ψ) ≡∫Rp

φ µn(t)ψ (t)dt

→∫Rp

φ µ (t)ψ (t)dt

= (2π)p/2 F−1 (µ)(ψ)

whenever ψ ∈S. Thus

µn (ψ) = FF−1µn (ψ)≡ F−1

µn (Fψ)→ F−1µ (Fψ)

≡ F−1Fµ (ψ) = µ (ψ).

The version of Lemma 59.18.4 is the following.

Lemma 59.19.4 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 59.19.2 the set, {µn}

∞

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

[x/∈Qr/2]|1−η | |ψ|dµn <

ε

3,

and ∫[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 59.19.3 because ψη ∈ S. This establishes theconclusion of the lemma in the case where ψ is also infinitely differentiable. To considerthe general case, let ψ only be uniformly continuous and let ψk = ψ ∗ φ k where φ k is a