**Unformatted text preview: **simple with Yn,k ↑ Xn and
E(Yn,k ) ↑ E(Xn ) as k → ∞ which is possible by Propositions 18.1 and 18.2. Set
Zk = max Yn,k ,
n ≤k and observe that 0 ≤ Zk ≤ Zk+1 so that (Zk ) is an increasing sequence of non-negative
simple random variables which necessarily has a limit
Z = lim Zk .
k→∞ By Proposition 18.2, E(Zk ) ↑ E(Z ). We now observe that if n ≤ k , then
Yn,k ≤ Zk ≤ Xn ≤ Xk ≤ X. (23.1) We now deduce from (23.1) that Xn ≤ Z ≤ X almost surely, and so letting n → ∞ implies
X = Z almost surely. We also deduce from (23.1) that
E(Yn,k ) ≤ E(Zk ) ≤ E(Xk )
for n ≤ k . Fix n and let k → ∞ to obtain
E(Xn ) ≤ E(Z ) ≤ lim E(Xk ).
k→∞ Now let n → ∞ to obtain
lim E(Xn ) ≤ E(Z ) ≤ lim E(Xk ). n→∞ k→∞ Thus,
E(Z ) = lim E(Xn ).
n→∞ But X = Z almost surely so that E(X ) = E(Z ) and we conclude
E(X ) = lim E(Xn )
n→∞ as required.
23–1 Theorem 23.2 (Fatou’s Lemma). Suppose that (Ω, F , P) is a probability space, and let
X1 , X2 , . . . be real-value...

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