Differentiation of Measures - Key Properties

The interplay between differentiation and integration in the Lebesgue setting reveals deep connections that generalize classical calculus results.

TheoremFundamental Theorem of Calculus for Lebesgue Integral (Part 1)

If $f \in L^1[a, b]$ and $F(x) = \int_a^x f(t) \, dt$ , then:

$F$ is absolutely continuous on $[a, b]$
$F$ is differentiable almost everywhere
$F'(x) = f(x)$ for almost every $x \in [a, b]$

This shows that indefinite integrals of $L^1$ functions are exactly the absolutely continuous functions whose derivatives equal the integrand almost everywhere.

TheoremFundamental Theorem of Calculus for Lebesgue Integral (Part 2)

If $F: [a, b] \to \mathbb{R}$ is absolutely continuous, then:

$F$ is differentiable almost everywhere
$F' \in L^1[a, b]$
$F(x) - F(a) = \int_a^x F'(t) \, dt$ for all $x \in [a, b]$

Thus every absolutely continuous function is an indefinite integral of its derivative.

ExampleFailure for Continuous Functions

The Cantor-Lebesgue function $F$ satisfies:

$F$ is continuous and increasing on $[0, 1]$
$F' = 0$ almost everywhere (since $F$ is constant on intervals totaling measure $1$ )
But $\int_0^1 F'(t) \, dt = 0 \neq 1 = F(1) - F(0)$

The failure occurs because $F$ is not absolutely continuous. The Fundamental Theorem requires absolute continuity, not just continuity.

Remark

Important properties and consequences:

Characterization of absolute continuity: $F$ is absolutely continuous if and only if $F(x) = F(a) + \int_a^x f(t) \, dt$ for some $f \in L^1[a, b]$ .
Chain rule: If $F$ is absolutely continuous and $\phi$ is Lipschitz, then $\phi \circ F$ is absolutely continuous with: $(\phi \circ F)'(x) = \phi'(F(x)) F'(x) \text{ a.e.}$
Integration by parts: For absolutely continuous $F$ and $G$ : $F(b)G(b) - F(a)G(a) = \int_a^b F'(x) G(x) \, dx + \int_a^b F(x) G'(x) \, dx$
Change of variables: If $\phi: [a, b] \to [c, d]$ is absolutely continuous and increasing, and $f \in L^1[c, d]$ : $\int_c^d f(y) \, dy = \int_a^b f(\phi(x)) \phi'(x) \, dx$
Substitution: This generalizes the classical substitution rule for definite integrals to the Lebesgue setting.

These results show that the Lebesgue integral provides the "right" setting for calculus, where differentiation and integration are true inverses under appropriate conditions.