Spaces of Random Variables

Abstract

This chapter is chiefly concerned with two metric spaces consisting of collections of random variables on a probability space (Ω, (ℱ, P): L₁(Ω, ℱ, P), consisting of all random variables X: Ω → ℝ such that E(|X|) < ∞, and L₂ (Ω, ℱ, P), consisting of those X for which E(X ²) < ∞. The space L₂(Ω, ℱ, P) has additional structure which makes it a ‘Hilbert space’. General Hilbert spaces are introduced in the first section, and L₂(Ω, ℱ, P) is treated in second section. Basic results from these two sections will play an important role in the definition of conditional probability distributions in Chapter 21. The metric space L₁(Ω, ℱ, P) is discussed briefly in the third section, and the final section of the chapter treats an application of Hilbert space methods to an estimation problem.