In probability theory, the conditional expectation is the expected value of a random variable given some information represented by a σ-algebra. The conditional expectation defined on the basis of a σ-algebra is again a random variable satisfying certain properties. In this chapter, we present the definition of conditional expectations based on this approach.

**Definition 14.1** (Conditional Expectation of a Random Variable Given a σ-Algebra). Let (Ω, , *P*) be a probability space, and *X* a random variable with *E*(|*X*|) < ∞. Let *G* be a sub-σ-algebra of . Then a version of the conditional expectation of *X* given *G* is defined to be a random variable *Y* (see Theorem 14.1) such that

(a) *Y* is *G*-measurable.

(b) *E*(|*Y*|) < ∞.

(c) For every *G* *G*, we have

The version of the conditional expectation of *X* given is written as *Y* = *E*(*X*|) a.s.

**Definition ...**

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