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