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

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

Statement

Let $f : X \to [0, + \infty]$ be measurable. Then $\int f d μ = 0 ⟺ f = 0 almost everywhere.$

Facts & Assumptions

Given: A nonnegative measurable function $f$ .

[L1]

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

[L2]

A statement holds almost everywhere when its exceptional set is contained in a measurable null set (Measure-null sets and almost-everywhere statements relative to a measure).

[L3]

The nonnegative integral is the supremum of the integrals of simple minorants (The nonnegative Lebesgue integral).

Proof

technique · direct

1.1

Assume $\int f d μ = 0$ . For $n \geq 1$ let $E_{n} : = {f \geq 1 / n}$ . Then[L1, L2, given, algebra] $(1 / n) χ_{E_{n}} \leq f$ , so [L1] gives $\frac{1}{n} μ (E_{n}) = \int (1 / n) χ_{E_{n}} d μ \leq \int f d μ = 0.$ Hence $μ (E_{n}) = 0$ for every $n$ . Since ${f > 0} = ⋃_{n} E_{n}$ , the exceptional set where $f \neq 0$ is null, so $f = 0$ almost everywhere by [L2].

1.2

Assume $f = 0$ almost everywhere, and let $N$ be a measurable null set[L2, L3, given] containing ${f > 0}$ . If $s = \sum_{j} c_{j} χ_{E_{j}}$ is a simple minorant of $f$ , then every set $E_{j}$ with $c_{j} > 0$ lies inside ${f > 0} \subseteq N$ , so $μ (E_{j}) = 0$ ; the remaining coefficients are $0$ . Therefore $\int s d μ = 0$ . Taking the supremum over all simple minorants in [L3] gives $\int f d μ = 0$ .

2.1

Step 1.1 proves the forward implication and step 1.2 proves the reverse [step 1.1, step 1.2] ∎ implication.

Depends on

Used by

Dependency tree · two levels

10 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, Proposition 8.1 (standard reference, not scraped)
Gerald B. Folland, Real Analysis, 2nd ed., Proposition 2.16 (standard reference, not scraped)

A nonnegative measurable function has integral 00 exactly when it vanishes almost everywhere