CorollaryStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

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A nonnegative measurable function with finite integral is finite almost everywhere

Statement

If $f : X \to [0, + \infty]$ is measurable and $\int f d μ < + \infty$ , then $f (x) < + \infty$ for almost every $x$ .

Facts & Assumptions

Given: A nonnegative measurable function $f$ with finite integral.

[L1]

The nonnegative integral is monotone and homogeneous (Monotonicity and nonnegative homogeneity of the nonnegative integral).

[L2]

A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere (A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere).

Proof

technique · direct

1.1

Let $F : = {f = + \infty}$ . Then $F \subseteq {f \geq n}$ for every $n \geq 1$ , so[L1, given] $n χ_{F} \leq f .$ By [L1], $n μ (F) = \int n χ_{F} d μ \leq \int f d μ < + \infty .$

2.1

If $μ (F) > 0$ , the inequality in step 1.1 would fail for large $n$ . [step 1.1, L2] ∎ Therefore $μ (F) = 0$ , so the indicator $χ_{F}$ has integral $0$ and hence vanishes almost everywhere by [L2]. Equivalently, $f < + \infty$ almost everywhere.

Depends on

Used by

Dominated convergence Theorem

Dependency tree · two levels

6 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

John K. Hunter, Measure Theory Notes, Proposition 4.14 (standard reference, not scraped)
Gerald B. Folland, Real Analysis, 2nd ed., Proposition 2.20 (standard reference, not scraped)