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**Unformatted text preview: **sitive random variable, then we deﬁne
E(X ) = sup{E(Y ) : Y is simple and 0 ≤ Y ≤ X }.
If X is any random variable, then we can write X = X + − X − where both X + and X − are
positive random variables. Provided that both E(X + ) and E(X − ) are ﬁnite, we deﬁne E(X )
to be
E(X ) = E(X + ) − E(X − )
and we say that X is integrable (or has ﬁnite expectation ).
Remark. If X : (Ω, F , P) → (R, B ) is a random variable, then we sometimes write
E(X ) =
X dP =
X ( ω ) d P {ω } =
X (ω )P {dω } .
Ω Ω Ω That is, the expectation of a random variable is the Lebesgue integral of X . We will say
more about this later.
Deﬁnition. Let L1 (Ω, F , P) be the set of real-valued random variables on (Ω, F , P) with
ﬁnite expectation. That is,
L1 (Ω, F , P) = {X : (Ω, F , P) → (R, B ) : X is a random variable with E(X ) < ∞}.
We will often write L1 for L1 (Ω, F , P) and suppress the dependence on the underlying
probability space.
Theorem 19.1. Supp...

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