770 CHAPTER 28. THE NORMAL DISTRIBUTION

Theorem 28.4.8 Let Λ = {µn}∞

n=1 be a sequence of probability measures definedon the Borel sets of E a Polish space. If Λ is tight then there exists a probability measureλ and a subsequence of {µn}

∞

n=1 , still denoted by {µn}∞

n=1 such that whenever φ is acontinuous bounded complex valued function defined on E,

limn→∞

∫φdµn =

∫φdλ .

Conversely, if µn converges weakly to λ , then {µn} is tight.

Proof: By tightness, there exists an increasing sequence of compact sets, {Kn} suchthat µ (Kn)> 1− 1

n for all µ ∈ Λ. Now letting µ ∈ Λ and φ ∈C (Kn) such that ∥φ∥∞≤ 1,

it follows ∣∣∣∣∫Kn

φdµ

∣∣∣∣≤ µ (Kn)≤ 1

and so the restrictions of the measures of Λ to Kn are contained in the unit ball of C (Kn)′ .

Recall from the Riesz representation theorem, the dual space of C (Kn) is a space of com-plex Borel measures. Theorem 21.5.5 on Page 557 implies the unit ball of C (Kn)

′ is weak∗ sequentially compact. This follows from the observation that C (Kn) is separable whichfollows easily from the Weierstrass approximation theorem. Recall from the Riesz repre-sentation theorem, the dual space of C (Kn) is a space of complex Borel measures. Theorem21.5.5 on Page 557 implies the unit ball of C (Kn)

′ is weak ∗ sequentially compact. Thisfollows from the observation that C (Kn) is separable which is proved in Corollary 28.4.7and leads to the fact that the unit ball in C (Kn)

′ is actually metrizable by Theorem 21.5.5on Page 557.

Thus the unit ball in C (Kn)′ is actually metrizable by Theorem 21.5.5 on Page 557.

Therefore, there exists a subsequence of Λ, {µ1k} such that their restrictions to K1 convergeweak ∗ to a measure, λ 1 ∈C (K1)

′. That is, for every φ ∈C (K1) ,

limk→∞

∫K1

φdµ1k =∫

K1

φdλ 1

By the same reasoning, there exists a further subsequence {µ2k} such that the restrictionsof these measures to K2 converge weak ∗ to a measure λ 2 ∈ C (K2)

′ etc. Continuing thisway,

µ11,µ12,µ13, · · · → Weak∗ in C (K1)′

µ21,µ22,µ23, · · · → Weak∗ in C (K2)′

µ31,µ32,µ33, · · · → Weak∗ in C (K3)′

...

Here the jth sequence is a subsequence of the ( j−1)th. Let λ n denote the measure inC (Kn)

′ to which the sequence {µnk}∞

k=1 converges weak ∗. Let {µn} ≡ {µnn} , the diag-onal sequence. Thus this sequence is ultimately a subsequence of every one of the abovesequences and so µn converges weak ∗ in C (Km)

′ to λ m for each m.Claim: For p > n, the restriction of λ p to the Borel sets of Kn equals λ n.

Proof of claim: Let H be a compact subset of Kn. Then there are sets, Vl open in Knwhich are decreasing and whose intersection equals H. This follows because this is a metric