59.18. THE CENTRAL LIMIT THEOREM 1921

and so, considered as elements of S∗,

F−1 (λ X) = φ X (·)(2π)−(p/2) ∈ L∞.

By the dominated convergence theorem

(2π)p/2 F−1 (λ Xn)(ψ) ≡∫Rp

φ Xn(t)ψ (t)dt

→∫Rp

φ X (t)ψ (t)dt

= (2π)p/2 F−1 (λ X)(ψ)

whenever ψ ∈S. Thus

λ Xn (ψ) = FF−1λ Xn (ψ)≡ F−1

λ Xn (Fψ)→ F−1λ X (Fψ)

≡ F−1Fλ X (ψ) = λ X (ψ).

This proves the lemma.

Lemma 59.18.4 If φ Xn(t)→ φ X (t) , then if ψ is any bounded uniformly continuous func-

tion,limn→∞

∫Rp

ψdλ Xn =∫Rp

ψdλ X.

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 and η (x) ∈ [0,1]for all x. By Lemma 59.18.2 the set, {λ Xn}

∞

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

[x/∈Qr/2]|1−η | |ψ|dλ Xn <

ε

3,

and ∫[x/∈Qr/2]

|1−η | |ψ|dλ X <ε

3.

Thus, ∣∣∣∣∫Rpψdλ Xn −

∫Rp

ψdλ X

∣∣∣∣≤ ∣∣∣∣∫Rpψdλ Xn −

∫Rp

ψηdλ Xn

∣∣∣∣+∣∣∣∣∫Rpψηdλ Xn −

∫Rp

ψηdλ X

∣∣∣∣+ ∣∣∣∣∫Rpψηdλ X−

∫Rp

ψdλ X

∣∣∣∣≤ 2ε

3+

∣∣∣∣∫Rpψηdλ Xn −

∫Rp

ψηdλ X

∣∣∣∣< ε

whenever n is large enough by Lemma 59.18.3 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 amollifier whose support is in (−(1/k) ,(1/k))p. Then ψk converges uniformly to ψ and sothe desired conclusion follows for ψ after a routine estimate. This proves the lemma.