= Martingale proof of the integral representation of a Lipschitz function
{title2=$f(x)-f(0)=\int_0^x g(u)\,du,\quad |g|\leq L$}
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 $|g|\leq L$ extend it to every endpoint by <continuity>. This proves <absolute continuity of a function> without presupposing a <derivative>.
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