851_lectures17_24

# This yields lim inf exn lim eyn e lim yn lim inf

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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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## This document was uploaded on 04/07/2014.

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