Fundamental theorem of calculus for Lebesgue integrals

Builds on Absolute continuity Lebesgue measure Integrable function Almost everywhere Derivative

Statement

Let $a < b$ be real numbers, and let integrals on $[a, b]$ be taken with respect to Lebesgue measure.

Let $g \colon [a, b] \to \R$ be integrable and let $F(x) = \int_a^x g(t) \, dt$ for $x \in [a, b]$ . Then $F$ is absolutely continuous, and $F$ is differentiable with $F'(x) = g(x)$ for almost every $x \in [a, b]$ .
Let $f \colon [a, b] \to \R$ be absolutely continuous. Then $f$ is differentiable at almost every point of $[a, b]$ , the derivative $f'$ is integrable on $[a, b]$ , and $f(x) = f(a) + \int_a^x f'(t) \, dt$ for every $x \in [a, b]$ .

Remarks

Part 1 follows from the Lebesgue differentiation theorem in dimension one; absolute continuity of $F$ comes from the absolute continuity of the integral. For part 2, an absolutely continuous function has bounded variation, so it is a difference of two increasing functions and is differentiable almost everywhere with $f' \in L^1$ . The function $h = f - f(a) - \int_a^x f'$ is then absolutely continuous with $h' = 0$ almost everywhere, and an absolutely continuous function with zero derivative almost everywhere is constant.

Example: $f(x) = \abs{x}$ on $[-1, 1]$ is Lipschitz, hence absolutely continuous. Its derivative is $-1$ on $(-1, 0)$ and $1$ on $(0, 1)$ , undefined at $0$ , and $\abs{x} = 1 + \int_{-1}^{x} f'(t) \, dt$ for every $x$ .

Absolute continuity cannot be weakened to continuity in part 2. The Cantor function $c$ is continuous and increasing on $[0, 1]$ with $c' = 0$ almost everywhere, but $c(1) - c(0) = 1 \ne 0 = \int_0^1 c'(t) \, dt$ .

Going deeper

For a student

The theorem shows that the absolutely continuous functions on $[a, b]$ are exactly the indefinite Lebesgue integrals. It is the one-dimensional prototype of Sobolev spaces: absolutely continuous functions are the functions in $W^{1,1}(a, b)$ , up to the choice of representative. In the language of measures, it says that the Lebesgue–Stieltjes measure $df$ of an absolutely continuous $f$ has density $f'$ with respect to Lebesgue measure.

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Absolute continuity concept
Almost everywhere concept
Borel sigma-algebra concept
Derivative concept
Euclidean space object
Integrable function concept
Outer measure concept
Lebesgue measure object
Fundamental theorem of calculus for Lebesgue integrals statement

Tag	`RX7JSHMP8Z`
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About	Absolute continuity
Uses	Absolute continuity, Lebesgue measure, Integrable function, Almost everywhere, Derivative
Changed	2 November 2026