Absolutely Continuous Functions

Absolute continuity is the property that distinguishes functions arising as Lebesgue integrals from the larger class of functions of bounded variation. The Radon-Nikodym theorem gives the precise measure-theoretic statement: $\nu \ll \mu$ if and only if $\nu$ has a density with respect to $\mu$.

Statement

A function $F : [a,b] \to \mathbb{R}$ is absolutely continuous if for every $\varepsilon > 0$ there exists $\delta > 0$ such that for any finite collection of disjoint subintervals $(a_i, b_i) \subset [a,b]$,

Fundamental theorem of Lebesgue integration: $F$ is absolutely continuous on $[a,b]$ if and only if there exists $f \in L^1([a,b])$ such that

In this case $F' = f$ almost everywhere.

Radon-Nikodym theorem (measure-theoretic form): If $\mu$ and $\nu$ are $\sigma$-finite measures and $\nu \ll \mu$ (meaning $\mu(A) = 0 \Rightarrow \nu(A) = 0$), then there exists a measurable function $f : X \to [0,\infty]$ such that $\nu = \mu$-with-density $f$, i.e.,

Visualization

Cantor function vs. an AC function — comparison:

The Cantor function $C : [0,1] \to [0,1]$ is continuous, monotone, $C' = 0$ a.e. (it is constant on each removed interval), yet $C(0) = 0$ and $C(1) = 1$. It is not absolutely continuous.

The AC function $A(x) = \int_0^x \mathbf{1}_{[0,1/2]}(t)\,dt$ satisfies $A' = \mathbf{1}_{[0,1/2]}$ a.e.

$x$	Cantor $C(x)$	$A(x) = \int_0^x \mathbf{1}_{[0,1/2]}$	$A'(x)$
$0$	$0$	$0$	$1$
$1/4$	$1/4$	$1/4$	$1$
$1/3$	$1/2$	$1/3$	$1$
$1/2$	$1/2$	$1/2$	jumps
$2/3$	$1/2$	$1/2$	$0$
$3/4$	$3/4$	$1/2$	$0$
$1$	$1$	$1/2$	$0$

The Cantor function "hides" its total variation on a set of measure zero (the Cantor set). AC functions cannot do this.

Why AC fails for Cantor: Take $n$ disjoint intervals of total length $\delta$ from the complement of the Cantor set. The function changes by at most $\delta$ total — fine. But intervals inside the Cantor set (measure zero, but uncountably many) account for the entire variation from $0$ to $1$.

Variation budget:   [ AC function: proportional to length ]
                    [ Cantor function: can be 1 with total length → 0 ]

Proof Sketch

1. (If $F = F(a) + \int_a^x f$): Given $\varepsilon$, set $\delta = \varepsilon / (2\|f\|_{L^1})$... wait, use uniform integrability of $f$: since $f \in L^1$, for any $\varepsilon$ there is $\delta$ with $\int_E |f| < \varepsilon$ whenever $\lambda(E) < \delta$. Then $\sum |F(b_i) - F(a_i)| \leq \int_{\bigcup (a_i,b_i)} |f| < \varepsilon$.

2. (Only if): If $F$ is AC, define $\nu(A) = \int_A dF$ (the Lebesgue-Stieltjes measure). Then AC implies $\nu \ll \lambda$ (Lebesgue measure). By Radon-Nikodym, $\nu = \lambda$-with-density $f$ for some $f \in L^1$, giving $F(x) - F(a) = \int_a^x f$.

3. Differentiating via the Lebesgue Differentiation TheoremLebesgue Differentiation Theorem$$\lim_{r \to 0} \frac{1}{|B_r(x)|} \int_{B_r(x)} f \, d\mu = f(x) \text{ a.e.}$$For locally integrable f, the ball averages of f converge to f(x) at almost every point xRead more → recovers $F' = f$ a.e.

Connections

The Radon-Nikodym theorem is the measure-theoretic backbone of conditional expectation in probability (see Bayes' TheoremBayes' Theorem$$P(A \mid B) = \frac{P(B \mid A)\, P(A)}{P(B)}$$Reversing conditional probability to update beliefs from evidenceRead more →) and of the Hahn decomposition. The characterisation of AC functions is the precise form of the Fundamental Theorem of CalculusFundamental Theorem of Calculus$$\int_a^b f'(x)\,dx = f(b) - f(a)$$Integration and differentiation are inverse operationsRead more → for Lebesgue integration. Singular functions like the Cantor function illustrate that Monotone Convergence TheoremMonotone Convergence Theorem$$f \text{ monotone},\; \sup_n f(n) < \infty \implies f(n) \to \sup_n f(n)$$A bounded monotone sequence of reals always converges — the supremum is the limitRead more → arguments can fail unless absolute continuity is verified.

import Mathlib

open MeasureTheory Measure

/-- Radon-Nikodym theorem: if ν ≪ μ (absolutely continuous), then ν has a
    measurable density (Radon-Nikodym derivative) with respect to μ. -/
theorem radon_nikodym {α : Type*} [MeasurableSpace α]
    {μ ν : Measure α} [SigmaFinite μ] [HaveLebesgueDecomposition ν μ]
    (h : ν ≪ μ) :
    ν = μ.withDensity (ν.rnDeriv μ) :=
  (withDensity_rnDeriv_eq ν μ h).symm