DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-27

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Integrable real and complex functions, and their integrals

Definition

Let $f : X \to R$ be measurable. Its positive and negative parts are $f^{+} : = \max {f, 0}, f^{-} : = \max {- f, 0};$ they are measurable by Closure properties of measurable functions used by the integral and satisfy $f = f^{+} - f^{-}$ and $∣ f ∣ = f^{+} + f^{-}$ .

The Lebesgue integral of a real measurable function is defined whenever at most one of $\int f^{+} d μ$ and $\int f^{-} d μ$ is $+ \infty$ , in which case $\int f d μ : = \int f^{+} d μ - \int f^{-} d μ .$ The function is integrable when both integrals are finite, equivalently when $\int ∣ f ∣ d μ < + \infty$ .

For a complex measurable function $h = u + i v$ with $u = Re h, v = Im h$ (Real and imaginary parts, complex conjugation, and modulus), define $h$ to be integrable when $∣ h ∣$ is integrable, and then define $\int h d μ : = \int u d μ + i \int v d μ .$

Depends on

Used by

A pointwise limit of integrable functions need not be integrable Counterexample
The class L¹(μ) of integrable functions Definition
Integrating against a Dirac measure is evaluation at the point Example
FALSE: pointwise limits of integrable functions are integrable False statement
The indefinite integral of an integrable function is countably additive on measurable sets Proposition
Absolute continuity of the integral Theorem
Dominated convergence Theorem
Integrable simple functions are dense in L¹(μ) Theorem
Jensen's integral inequality for a probability measure Theorem
The Lebesgue integral is linear on L¹(μ) Theorem
The modulus of an integral is bounded by the integral of the modulus Theorem
Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree Theorem

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Richard F. Bass, Real Analysis for Graduate Students, Definition 6.2 (standard reference, not scraped)
John K. Hunter, Measure Theory Notes, Definition 4.8 (standard reference, not scraped)