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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 )
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 .
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