**Unformatted text preview: **ose that (Ω, F , P) is a probability space, and let X1 , X2 , . . ., X , and Y
all be real-valued random variables on (Ω, F , P).
(a) L1 is a vector space and expectation is a linear map on L1 . Furthermore, expectation
is positive. That is, if X , Y ∈ L1 with 0 ≤ X ≤ Y , then 0 ≤ E(X ) ≤ E(Y ).
19–1 (b) X ∈ L1 if and only if |X | L1 , in which case we have
|E(X )| ≤ E(|X |).
(c) If X = Y almost surely (i.e., if P {ω : X (ω ) = Y (ω )} = 1), then E(X ) = E(Y ).
(d) (Monotone Convergence Theorem) If the random variables 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→∞ (We allow E(X ) = +∞ if necessary.) (e) (Fatou’s Lemma) If the random variables Xn all satisfy Xn ≥ Y almost surely for some
Y ∈ L1 and for all n, then
E lim inf Xn ≤ lim inf E(Xn ).
(19.1)
n→∞ n→∞ In particular, (19.1) holds if Xn ≥ 0 for all n. (f ) (Lebesgue’s Dominated Convergence Theorem) If the random variables Xn → X , and
if for some Y ∈ L1 we have |Xn | ≤ Y almost...

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