851_lectures17_24

# 201 theorem 202 if x f p r b is a random variable

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Unformatted text preview: and let α ∈ [0, ∞). We know that there exist sequences Xn and Yn of non-negative simple random variables such that (i) Xn ↑ X and E(Xn ) ↑ E(X ), and (ii) Yn ↑ Y and E(Yn ) ↑ E(Y ). Therefore, αXn is a sequence of non-negative simple random variables with (αXn ) ↑ (αX ) and E(αXn ) ↑ E(αXn ). Moreover, Xn + Yn is a sequence of non-negative simple random variables with (Xn + Yn ) ↑ (X + Y ) and E(Xn + Yn ) ↑ E(X + Y ). However, we know that expectation is linear on simple random variables so that E(αXn ) = αE(Xn ) and E(Xn + Yn ) = E(Xn ) + E(Yn ). Thus, we ﬁnd E(α X n ) = α E(Xn ) E ( X n + Yn ) = E ( X n ) + E ( Y n ) ↓ ↓ ↓ ↓ and E(α X ) α E(X ) E(X + Y ) E(X ) + E(Y ) so by uniqueness of limits, we conclude E(αX ) = αE(X ) and E(X + Y ) = E(X + Y ). Also note that if 0 ≤ X ≤ Y , then the deﬁnition of expectation of non-negative random variables immediately implies that 0 ≤ E(X ) ≤ E(Y ) so that Y ∈ L1 implies X ∈ L1 . Suppose now...
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