Bounded variation and absolute continuity. Let \(\lambda\) be one-dimensional Lebesgue measure.

A function \(F:[a, b] ightarrow \mathbb{R}\) is said to be

absolutely continuous (AC) if for each \(\epsilon>0\) there is a \(\delta>0\) such that \(\sum_{n=1}^{N}\left|F\left(y_{n}ight)-F\left(x_{n}ight)ight| \leqslant \epsilon \quad\) for \(\quad\) all \(\quad a \leqslant x_{1}

\(\sup \sum_{n=1}^{N}\left|F\left(t_{n}ight)-F\left(t_{n-1}ight)ight|\) is finite, where the sup extends over all finite partitions \(t_{0}=a(i) We have \(\mathrm{AC}[a, b] \subset C[a, b] \cap \mathrm{BV}[a, b]\).
(ii) Any \(F \in \mathrm{BV}[a, b]\) can be written as a sum of increasing functions \(f\) and \(g\).
[\(\operatorname{try} f(t)=V(f,[a, t])\).]
(iii) \(F(x):=\int_{[a, x)} f(t) \lambda(d t)\) is \(\mathrm{AC}\) for all \(f \in \mathcal{L}^{1}(\mu)\).
(iv) If \(F\) is \(\mathrm{AC}\), there is some \(\phi \in \mathcal{L}^{1}(\lambda)\) such that \(F(x)=\int_{(-\infty, x)} \phi(t) \lambda(d t)\).
[ decompose \(F=f_{1}-f_{2}\) with \(f_{i}\) increasing and AC.]