851_lectures17_24

# A l1 is a vector space and expectation is a linear

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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 ) &lt; ∞}. 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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## This document was uploaded on 04/07/2014.

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