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**Unformatted text preview: **ons is measurable.
Theorem 22.3. Suppose that (W, F ), (X, G ), and (Y, H) are measurable spaces, and let
f : (W, F ) → (X, G ) and g : (X, G ) → (Y, H) be measurable. Then the function h = g ◦ f
is a measurable function from (W, F ) to (Y, H).
Proof. Suppose that H ∈ H. Since g is measurable, we have g −1 (H ) ∈ G . Since f is
measurable, we have f −1 (g −1 (H )) ∈ F . Since
h−1 (H ) = (g ◦ f )−1 (H ) = f −1 (g −1 (H )) ∈ F
the proof is complete. 22–3 Statistics 851 (Fall 2013)
Prof. Michael Kozdron October 30, 2013 Lecture #23: Proofs of the Main Expectation Theorems
(continued)
Theorem 23.1 (Monotone Convergence Theorem). Suppose that (Ω, F , P) is a probability
space, and let X1 , X2 , . . ., and X be real-valued random variables on (Ω, F , P). If Xn ≥ 0
for all n and Xn ↑ X (i.e., Xn → X and Xn ≤ Xn+1 ), then
lim E(Xn ) = E lim Xn = E(X )
n→∞ n→∞ (allowing E(X ) = +∞ if necessary). Proof. For every n, let Yn,k , k = 1, 2, . . ., be non-negative and...

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