61.4. A MAJOR EXISTENCE AND CONVERGENCE THEOREM 1995

from 61.3.7,

λ(KC

n)

= limp→∞

λ p(KC

n ∩Kp)

≤ lim supp→∞

(λ p (Kp)−λ p (Kn))

≤ lim supp→∞

(λ p (Kp)−λ n (Kn))

≤ lim supp→∞

(1−(

1− 1n

))=

1n

Suppose ||φ ||∞< M. Then∣∣∣∣∫ φdµk−

∫φdλ

∣∣∣∣≤ ∣∣∣∣∫KCn

φdµk +∫

Kn

φdµk−(∫

Kn

φdλ +∫

KCn

φdλ

)∣∣∣∣≤∣∣∣∣∫Kn

φdµk−∫

Kn

φdλ n

∣∣∣∣+ ∣∣∣∣∫KCn

φdµk−∫

KCn

φdλ

∣∣∣∣≤

∣∣∣∣∫Kn

φdµk−∫

Kn

φdλ n

∣∣∣∣+ ∣∣∣∣∫KCn

φdµk

∣∣∣∣+ ∣∣∣∣∫KCn

φdλ

∣∣∣∣≤

∣∣∣∣∫Kn

φdµk−∫

Kn

φdλ n

∣∣∣∣+ Mn+

Mn

First let n be so large that 2M/n < ε/2 and then pick k large enough that the above expres-sion is less than ε.

Definition 61.3.6 Let E be a complete separable metric space and let µ and the sequenceof probability measures, {µn} defined on B (E) satisfy

limn→∞

∫φdµn =

∫φdµ.

for every φ a bounded continuous function. Then µn is said to converge weakly to µ .

61.4 A Major Existence And Convergence TheoremHere is an interesting lemma about weak convergence.

Lemma 61.4.1 Let µn converge weakly to µ and let U be an open set with µ (∂U) = 0.Then

limn→∞

µn (U) = µ (U) .

Proof: Let {ψk} be a sequence of bounded continuous functions which decrease toXU . Also let {φ k} be a sequence of bounded continuous functions which increase to XU .For example, you could let

ψk (x) ≡ (1− k dist(x,U))+ ,

φ k (x) ≡ 1−(1− k dist

(x,UC))+ .