Lebesgue's differentiation theorem for monotone functions

Let $F:[a,b]\to \mathbb{R}$ be monotone. Then $F$ is differentiable at almost every point of $[a,b]$, the derivative $F^{\mathrm{\prime }}$ is measurable, and if $F$ is increasing then

${\int }_a^b{F}^{\mathrm{\prime }}d\lambda \le F(b)-F(a),$

with equality precisely when $F$ is absolutely continuous. The same conclusion holds for every $F$ of bounded variation, since such an $F$ is a difference of two increasing functions.

The inequality is genuinely an inequality. The Cantor function $c:[0,1]\to [0,1]$ is continuous and increasing with $c^{\mathrm{\prime }}=0$ almost everywhere, so ${\int }_0^1{c}^{\mathrm{\prime }}d\lambda =0<1=c(1)-c(0)$.

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

What would prove it. The Vitali covering theorem (Vitali covering theorem ‡) applied to the sets where the upper and lower Dini derivates differ, or the rising sun lemma route, or the mini-Vitali route through growth lemmas; all three need Lebesgue outer measure as a measure, not just the elementary null sets. The inequality ${\int }_a^b{F}^{\mathrm{\prime }}\le F(b)-F(a)$ then follows from Fatou's lemma (Fatou's lemma ‡) applied to the difference quotients $n(F(x+1\mathrm{/}n)-F(x))$.

The equality case. Equality ${\int }_a^b{F}^{\mathrm{\prime }}d\lambda =F(b)-F(a)$ holds for an increasing $F$ exactly when $F$ is absolutely continuous in the sense of Absolutely continuous functions ‡, which is the content of The sharp fundamental theorem of calculus (absolute continuity) ‡. The gap between the two sides is carried by the singular part of $F$, and the Cantor function is the case where the singular part is everything.

Which page it serves. The monotone functions and discontinuities page, which proves that a monotone function has at most countably many discontinuities and therefore is continuous almost everywhere, and then stops. Differentiability almost everywhere is the next statement in every classical treatment and cannot be reached from the elementary theory. It is also what allows the Cantor function counterexample to be stated at full strength on the Cantor set page: not merely "$c^{\mathrm{\prime }}=0$ off a null set", which is elementary, but "$c$ is one of the monotone functions to which Lebesgue's theorem applies, and it saturates the inequality in the wrong direction".

A naming warning. The phrase "Lebesgue differentiation theorem" is used for two different results. In the classical one-variable literature, including Thomson's article cited above, it names the theorem stated here, that monotone functions are almost everywhere differentiable. In the $L^1$ and harmonic analysis literature it names the averaging statement recorded separately as Lebesgue differentiation theorem for $L^1$ functions ‡. Neither is proved here, and this library keeps them under distinct names.