Kenneth Kuttler

59.19. CHARACTERISTIC FUNCTIONS, PROKHOROV THEOREM 1929

mollifier 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.

The next theorem is really important. It gives the existence of a measure based on anassumption that a set of measures is tight. The next theorem is Prokhorov’s theorem abouta tight set of measures. Recall that Λ is tight means that for every ε > 0 there exists Kcompact such that µ

(KC)< ε for all µ ∈ Λ.

Theorem 59.19.5 Let Λ = {µn}∞

n=1 be a sequence of probability measures defined on theBorel sets of Rp. If Λ is tight then there exists a probability measure, λ and a subsequenceof {µn}

∞

n=1 , still denoted by {µn}∞

n=1 such that whenever φ is a continuous bounded com-plex valued function defined on E,

limn→∞

∫φdµn =

∫φdλ .

Proof: By tightness, there exists an increasing sequence of compact sets, {Kn} suchthat

µ (Kn)> 1− 1n

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 17.5.5 on Page 462 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. Thus the unit ball in C (Kn)

′ isactually metrizable by Theorem 17.5.5 on Page 462. Therefore, there exists a subsequenceof Λ, {µ1k} such that their restrictions to K1 converge weak ∗ to a measure, λ 1 ∈C (K1)

′.That is, for every φ ∈C (K1) ,

limk→∞

∫K1

φdµ1k =∫

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.

59.19. CHARACTERISTIC FUNCTIONS, PROKHOROV THEOREM 1929mollifier whose support is in (— (1/k) ,(1/k))”. Then y;, converges uniformly to y and sothe desired conclusion follows for y after a routine estimate. fjThe next theorem is really important. It gives the existence of a measure based on anassumption that a set of measures is tight. The next theorem is Prokhorov’s theorem abouta tight set of measures. Recall that A is tight means that for every € > 0 there exists Kcompact such that pt (K©) < é for all pp € A.Theorem 59.19.5 Let A= {L,},,_, be a sequence of probability measures defined on theBorel sets of R”. If A is tight then there exists a probability measure, A and a subsequenceof {L,};,—1 » still denoted by {1}; such that whenever @ is a continuous bounded com-plex valued function defined on E,lim [ gan, = [ oda.Proof: By tightness, there exists an increasing sequence of compact sets, {K,} suchthat1K,)>1—-—LU (Kn) > ifor all u € A. Now letting € A and @ € C(K,) such that ||@||,, < 1, it followsLL ody| <M(Kn) <1and so the restrictions of the measures of A to K, are contained in the unit ball of C(K,)'.Recall from the Riesz representation theorem, the dual space of C(K,) is a space of com-plex Borel measures. Theorem 17.5.5 on Page 462 implies the unit ball of C(K,)’ is weak* sequentially compact. This follows from the observation that C(K,,) is separable whichfollows easily from the Weierstrass approximation theorem. Thus the unit ball in C (K,,)’ isactually metrizable by Theorem 17.5.5 on Page 462. Therefore, there exists a subsequenceof A, {u,,} such that their restrictions to K; converge weak * to a measure, 2; € C(K)’.That is, for every @ € C(K1),lim | bduye= | odhyk-y00 J Ky KyBy the same reasoning, there exists a further subsequence {j15,} such that the restrictionsof these measures to Ky converge weak * to a measure Ap €C (K2)' etc. Continuing thisway,My sHy2,Hy3.-77 + Weak» in C(Ki)!Lp) Hp, H23,7> + Weak in C(K3)'[31 HL3,H33,-7* + Weak in C(K3)'Here the j’” sequence is a subsequence of the (j— 1)" . Let A, denote the measure inC(Kn)’ to which the sequence {,,,};_, converges weak *. Let {u,,} = {lyn}, the diag-onal sequence. Thus this sequence is ultimately a subsequence of every one of the abovesequences and so LL, converges weak * in C(Km)’ to Am for each m.