Generalized Radon-Nikodym Derivatives

Abstract

Let (Ω, B, μ)be any probability space and let T: Ω → Ω be a measurable mapping. If w ↦ f (w) is a measurable function on Ω, the composition f ∘ T is also a measurable function on Ω. Denote by {f _α , α ∈ A} the μ-equivalence class of random variables on (Ω, B, μ) associated to f, i.e., for every α ∈ A, μ{w: f _α (w) ≠ f(w)}= 0. The question whether the mapping f ↦ f ∘ T is well-defined on the μ-equivalence classes is of paramount importance in probability theory. It is easily verified that the answer to this question is affirmative if and only if T* μ ≪ μ. However, in case this absolute continuity property is not satisfied but the equivalence class {f _α , α ∈ A} has a “distinguished member”, say f _o then we are free to define the equivalence class f ∘ T as the equivalence class corresponding to f _o ∘ T.