Martingale proof of the integral representation of a Lipschitz function ($f(x)-f(0)={\int }_0^{x}g(u)du,|g|\le L$)

For an $L$-Lipschitz function $f$ on $[0,1]$, its slopes on the dyadic partition, evaluated at a uniform distribution input, form a bounded martingale for the dyadic filtration. The uniformly integrable martingale convergence theorem yields a bounded terminal random variable $g(X)$, with $g$ chosen as a Borel measurable function by the Doob-Dynkin lemma. The terminal conditional expectation identity says that each slope is the average of $g$ over the corresponding dyadic interval. Integrating over adjacent dyadic intervals and taking a telescoping sum proves the displayed representation first at dyadic endpoints. The Lipschitz condition for $f$ and the bound $|g|\le L$ extend it to every endpoint by continuity. This proves absolute continuity of a function without presupposing a derivative.