= Solution
{c}
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 $h_n(f(x))=k2^{-n}$. The chosen interval is open on the left and closed on the right, so
$$
\boxed{0<g(x)-h_n(f(x))\leq2^{-n}.}
$$
This 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 gives
$$
\boxed{g=h\circ f.}
$$
This is the factorization conclusion of the <Doob-Dynkin lemma> under the specified Borel-image hypothesis.
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