 Solution

For $x\in M$, let $k$ be the unique index such that $g(x)\in B_{n,k}$. Then $f(x)\in A_{n,k}$ and $h_n(f(x))=k{2}^{-n}$. The chosen interval is open on the left and closed on the right, soThis remains correct at dyadic endpoints; changing it to a floor approximation with a non-strict left endpoint would describe a different approximation.

By part (iii), define the finite limit $h(t)={\lim }_n{h}_n(t)$ for every $t\in \mathbb{R}$. Part (a) makes this limit Borel measurable. Letting $n\to \infty$ in the displayed error bound givesThis is the factorization conclusion of the Doob-Dynkin lemma under the specified Borel-image hypothesis.