Differentiation of Measures - Main Theorem

TheoremLebesgue's Differentiation Theorem for Monotone Functions

Let $F: [a, b] \to \mathbb{R}$ be a monotone increasing function. Then:

$F$ is differentiable almost everywhere on $[a, b]$
$F' \in L^1[a, b]$
$\int_a^b F'(x) \, dx \leq F(b) - F(a)$

Equality holds in (3) if and only if $F$ is absolutely continuous.

Moreover, if $F$ is absolutely continuous, then: $F(x) - F(a) = \int_a^x F'(t) \, dt \text{ for all } x \in [a, b]$

This fundamental result shows that monotone functions are differentiable almost everywhere, a remarkable regularity property. The inequality in (3) becomes an equality precisely when $F$ has no "singular" part in its growth.

ExampleStrict Inequality Case

For the Cantor-Lebesgue function $C: [0, 1] \to [0, 1]$ :

$C$ is continuous and increasing
$C'(x) = 0$ for almost every $x$ (on the complement of the Cantor set)
$\int_0^1 C'(x) \, dx = 0 < 1 = C(1) - C(0)$

The strict inequality reflects the singular continuous nature of $C$ : it increases without having a positive derivative almost anywhere.

Remark

Important consequences and extensions:

Bounded variation functions: Every function of bounded variation is differentiable a.e., since it can be written as the difference of two monotone functions.
Lebesgue-Stieltjes measures: For increasing $F$ , define a measure $\mu_F$ by $\mu_F([a, b)) = F(b) - F(a)$ . The theorem shows: $\frac{d\mu_F}{dx}(x) = F'(x) \text{ a.e.}$

The Lebesgue decomposition splits $\mu_F$ into absolutely continuous and singular parts.

Functions of several variables: The theorem extends to functions on $\mathbb{R}^n$ . For $F: \mathbb{R}^n \to \mathbb{R}$ of bounded variation, partial derivatives exist almost everywhere.
Vitali covering theorem: The proof relies on the Vitali covering lemma, which is a fundamental tool in geometric measure theory.
Rising sun lemma: An alternative proof uses the "rising sun lemma," a geometric argument involving the "shadow" cast by the graph of a function.

This theorem is one of Lebesgue's crowning achievements, revealing deep structure in the class of monotone functions and connecting differentiation with measure theory.