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**Unformatted text preview: **ne E(X ) for positive random variables,
(3) deﬁne E(X ) for general random variables.
This strategy is sometimes called the “standard machine” and is the outline that we will
follow to prove most results about expectation of random variables.
Step 1: Simple Random Variables
Let (Ω, F , P) be a probability space. Suppose that X : Ω → R is a simple random variable
so that
m
aj 1 A j ( ω )
X (ω ) =
j =1 where a1 , . . . , am ∈ R and A1 , . . . , Am ∈ F . We deﬁne the expectation of X to be
E(X ) = m
j =1 aj P {Aj } . Step 2: Positive Random Variables
Suppose that X is a positive random variable. That is, X (ω ) ≥ 0 for all ω ∈ Ω. (We will
need to allow X (ω ) ∈ [0, +∞] for some consistency.) We are also assuming at this step that
X is not a simple random variable.
Deﬁnition. If X is a positive random variable, deﬁne the expectation of X to be
E(X ) = sup{E(Y ) : Y is simple and 0 ≤ Y ≤ X }. (18.1) Proposition 18.1. For every random variable X ≥ 0, there exists a sequence (Xn ) of
positive, simple random variables with Xn...

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