Differentiation of Measures - Core Definitions

The differentiation of measures extends classical calculus notions to the measure-theoretic setting, connecting integration with differentiation through the Fundamental Theorem of Calculus for Lebesgue integrals.

DefinitionAbsolutely Continuous Function

A function $F: [a, b] \to \mathbb{R}$ is absolutely continuous if for every $\epsilon > 0$ , there exists $\delta > 0$ such that for any finite collection of disjoint intervals $\{(a_i, b_i)\}_{i=1}^n$ with $\sum_{i=1}^n (b_i - a_i) < \delta$ : $\sum_{i=1}^n |F(b_i) - F(a_i)| < \epsilon$

Absolutely continuous functions are uniformly continuous, but the converse is false. They are the natural class of functions that arise as indefinite integrals.

ExampleAbsolutely Continuous vs. Continuous

Absolutely continuous: $F(x) = x^2$ on $[0, 1]$ is absolutely continuous (and differentiable everywhere).
Continuous but not absolutely continuous: The Cantor-Lebesgue function (Devil's staircase) is continuous, increasing, and maps $[0, 1]$ onto $[0, 1]$ , but is not absolutely continuous. It is constant on the complement of the Cantor set, which has measure $1$ , yet increases from $0$ to $1$ .
Not continuous: $F(x) = \begin{cases} 0 & x < 0 \\ 1 & x \geq 0 \end{cases}$ is not continuous, hence not absolutely continuous.

DefinitionFunction of Bounded Variation

A function $F: [a, b] \to \mathbb{R}$ has bounded variation if: $V_a^b(F) = \sup \left\{\sum_{i=1}^n |F(x_i) - F(x_{i-1})| : a = x_0 < x_1 < \cdots < x_n = b\right\} < \infty$

The supremum $V_a^b(F)$ is called the total variation of $F$ on $[a, b]$ .

Functions of bounded variation can be written as the difference of two increasing functions: $F = F_1 - F_2$ .

Remark

Relationships between function classes:

Absolutely continuous $\Rightarrow$ bounded variation: Every absolutely continuous function on $[a, b]$ has bounded variation.
Absolutely continuous $\Rightarrow$ continuous: Absolute continuity implies ordinary continuity.
Bounded variation $\not\Rightarrow$ continuous: Functions of bounded variation can have jump discontinuities.
Differentiable a.e.: Functions of bounded variation are differentiable almost everywhere (by a theorem of Lebesgue).
Indefinite integrals: If $f \in L^1[a, b]$ , then $F(x) = \int_a^x f(t) \, dt$ is absolutely continuous, and $F'(x) = f(x)$ for almost every $x$ .

These definitions provide the framework for understanding when functions can be recovered from their derivatives via integration, generalizing the classical Fundamental Theorem of Calculus.