MATH 418 : Lectures on Conditional Expectation

Conditional expectation is one of the most useful tools of probability. The Radon-Nikodym theorem enables us to construct conditional expectations. Definition 1. A measure µ on (Ω, F) is σ-finite iff there is A n ∈ F satisfying A n ↑ Ω where µ(A n) < ∞ for all n. Definition 2. Let µ, ν be measures on (Ω, F). Then ν is absolutely continuous with respect to µ (ν << µ) iff for all A ∈ F, µ(A) = 0 means ν(A) = 0.