Stochastic Calculus HW #1

Proof. By composing Y with a homeomorphism R→ (0, 1) if required, we may assume that Y is bounded and we do so. To prove the claim, we appeal to the standard machinery using the monotone class theorem. Let L∞ be the family of bounded σ(X )-measurable functions and let H be the subset of L∞ defined by H = {Y ∈ L∞ : ∃ a Borel-measurable f satisfying Y = f (X ) a.s.}. We claim thatH = L∞. To this end, it suffices to prove thatH satisfies the condition of (function-space version of) the monotone class theorem. That is, it is sufficient to establish the followings: (1) H is a vector space. (2) H contains any σ(X )-measurable indicator functions. (3) H is closed under monotone limit in the following sense: if (Yk ) is a sequence in H such that 0 ≤ Y1 ≤ Y2 ≤ · · · and Yk ↑ Y for some Y ∈ L∞, then Y ∈ H. The condition (1) is obvious. For (2), notice that any σ(X )-measurable set can be written as {X ∈ B} for some Borel set B. Thus if Y is an indicator function in L∞, then the event {Y = 1} can be written in this form and