= Dyadic slope-tail criterion for absolute continuity
For a <continuous function> $f$ on $[0,1]$, let $G_n$ be its <dyadic slope martingale>. Then $f$ is an <absolutely continuous function> if and only if
$$
\lim_{\lambda\to\infty}\sup_n\int_0^1|G_n(t)|\mathbf1_{\{|G_n(t)|\geq\lambda\}}dt=0.
$$
This is exactly <uniform integrability>. The <uniformly integrable martingale convergence theorem> gives <convergence in L1> $G_n\to g$, while their integrated <linear interpolations> converge uniformly to $f$. Hence $f(x)=f(0)+\int_0^xg$. Conversely, if $f$ has density $g\in L^1$, its slopes are $\mathbb E[g\mid\mathcal F_n]$, and the <uniform integrability of conditional expectations> proves the criterion. The dyadic tail <integral> is also the sum of the absolute endpoint increments in cells whose slope is at least $\lambda$ in magnitude.
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