Banach-Zarecki theorem

Let $F : [a,b] \to \mathbb{R}$. Then $F$ is absolutely continuous on $[a,b]$ if and only if all three of the following hold:

1. $F$ is continuous on $[a,b]$;
2. $F$ is of bounded variation on $[a,b]$;
3. $F$ has Luzin's property (N): $\lambda(F(E)) = 0$ for every $E \subseteq [a,b]$ with $\lambda(E) = 0$.

None of the three may be dropped. The Cantor function satisfies 1 and 2 and fails 3, since it maps the Cantor set, a null set, onto $[0,1]$. The function $x \mapsto x \sin(1/x)$ on $(0,1]$ extended by $0$ satisfies 1 and 3 and fails 2. A jump function satisfies 2 and 3 and fails 1.

Not proved in this library. It is recorded with citations and used in no proof here.

What would prove it. The forward direction is a covering estimate directly from the definition of absolute continuity (Absolutely continuous functions ‡). The converse uses the Vitali covering theorem (Vitali covering theorem ‡) together with the Banach indicatrix formula, which expresses the total variation of a continuous function as the integral over $\mathbb{R}$ of the number of preimages, and hence needs the Lebesgue integral (Lebesgue measure and the Lebesgue integral ‡).

Which page it serves. The bounded variation and Riemann-Stieltjes page. That page has both the continuity and the bounded variation hypotheses available as proved notions, and property (N) can be stated with the elementary covering notion of a null set, so the statement is fully intelligible there. Only the proof is out of reach.

Why it is worth recording even unproved. It is the answer to the question that the sharp fundamental theorem of calculus (The sharp fundamental theorem of calculus (absolute continuity) ‡) leaves open, namely what absolute continuity is intrinsically, with no integral in sight. The pair of statements together explains the Cantor function completely: it fails Newton-Leibniz because it fails property (N), and it fails property (N) because it moves a null set onto a set of full measure.