Let (S, F, μ) be a measure space. Here S is a set, F is a σ-algebra, µ is a σ-finite measure on F. If F0 is a σ-subalgebra of F, x∈L1 ((S, F, μ), then denote by EFo the unique, up to equivalence, F0-measurable function satisfying
for each A ∈ F0. By the Radon — Nikodym theorem, such function exists. The function EFox =EFo,μx is called the conditional expectation with respect to F0.