59.19. CHARACTERISTIC FUNCTIONS, PROKHOROV THEOREM 1931

the last equation holding by the monotone convergence theorem.It remains to verify

limk→∞

∫φdµk =

∫φdλ

for every φ bounded and continuous. This is where tightness is used again. Suppose||φ ||

∞< M. Then as noted above,

λ n (Kn) = λ (Kn)

because for p > n,λ p (Kn) = λ n (Kn) and so letting p→ ∞, the above is obtained. Also,from 59.19.47,

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

Consequently,∣∣∣∣∫ 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 59.19.6 Let µ,{µn} be probability measures defined on the Borel sets of Rp

and let the sequence of probability measures, {µn} satisfy

limn→∞

∫φdµn =

∫φdµ.

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

With the above, it is possible to prove the following amazing theorem of Levy.