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**Unformatted text preview: **that X and Y are general random variables and let α ∈ R. Since
( α X ) = (α X ) + − ( α X ) − ≤ ( α X ) + + ( α X ) − ≤ | α | ( X + + X ) and
(X + Y ) = ( X + Y )+ − (X + Y )− ≤ (X + Y )+ + ( X + Y )− ≤ X + + X − + Y + + Y − . Hence, since X + , X − , Y + , Y − ∈ L1 , we conclude that αX ∈ L1 and X + Y ∈ L1 . Finally,
since E(X ) = E(X + ) − E(X − ) by deﬁnition, and since we showed above that expectation is
linear on non-negative random variables, we conclude that expectation is linear on general
random variables.
20–1 Theorem 20.2. If X : (Ω, F , P) → (R, B ) is a random variable, then X ∈ L1 if and only if
|X | ∈ L 1 .
Proof. Suppose that X ∈ L1 so that E(X + ) < ∞ and E(X − ) < ∞. Since |X | = X + + X −
and since expectation is linear, we conclude that
E(|X |) = E(X + + X − ) = E(X + ) + E(X − ) < ∞
so that |X | ∈ L1 . On the other hand, suppose that |X | ∈ L1 so that E(X + ) + E(X − ) < ∞. However, since
X + ≥ 0 and X −...

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