Conditional Expectation - Basic Properties

Basic Properties

Let (Ω, M, P) be a probability space, and let N be a σ-subalgebra of M.

• Conditioning with respect to N  is linear on the space of integrable real random variables.

• More generally, for every integrable N–measurable random variable Y on Ω.

•   for all B ∈ N and every integrable random variable X on Ω.

• Jensen's inequality holds: If ƒ is a convex function, then

• Conditioning is a contractive projection

for any s ≥ 1.