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

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Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree

Statement

Let $f, g \in L^{1} (μ)$ . Then the following are equivalent:

$f = g$ almost everywhere;
for every measurable $A$ , $\int_{A} f d μ = \int_{A} g d μ .$

For integrable real or complex $h$ , the notation in condition 2 means $\int_{A} h d μ : = \int h χ_{A} d μ$ ; the product is integrable because $∣ h χ_{A} ∣ \leq ∣ h ∣$ .

Facts & Assumptions

Given: Integrable functions $f, g \in L^{1} (μ)$ .

[L1]

The integral over a null set vanishes for nonnegative integrands (A nonnegative integral over a null set vanishes).

[L2]

The Lebesgue integral is linear on $L^{1} (μ)$ (The Lebesgue integral is linear on $L^{1} (μ)$ ).

[L3]

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).

[L4]

Real and imaginary parts of an integrable complex function are integrable (Integrable real and complex functions, and their integrals).

Proof

technique · direct

1.1

Assume $f = g$ almost everywhere, with exceptional null set $N$ . Then for [L1, L2, L4, given] every measurable $A$ , the real and imaginary parts of $(f - g) χ_{A}$ are supported on $N$ , so [L1] and [L2] give $\int_{A} (f - g) d μ = 0,$ hence $\int_{A} f d μ = \int_{A} g d μ$ .

1.2

Assume instead that $\int_{A} f d μ = \int_{A} g d μ$ for every measurable [L2, L3, L4, given] $A$ . Apply this to the real part $u : = Re (f - g)$ on the set $A_{+} : = {u > 0}$ and to $- u$ on $A_{-} : = {u < 0}$ . In each case the corresponding nonnegative integral is $0$ , so [L3] gives $u = 0$ almost everywhere. The same argument for $v : = Im (f - g)$ shows $v = 0$ almost everywhere. Hence $f = g$ almost everywhere.

2.1

Step 1.1 proves $(1) \Rightarrow (2)$ and step 1.2 proves $(2) \Rightarrow (1)$ .

step 1.1step 1.2∎

Depends on

Used by

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Dependency tree · two levels

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Sources

Richard F. Bass, Real Analysis for Graduate Students, Proposition 8.2 (standard reference, not scraped)
Gerald B. Folland, Real Analysis, 2nd ed., Proposition 2.23(b) (standard reference, not scraped)