False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-27

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FALSE: the Lebesgue integral extends linearly to all measurable functions

Statement

Whenever measurable real-valued functions $f$ , $g$ , and $f + g$ all have defined extended Lebesgue integrals, the extended-real sum $\int f d μ + \int g d μ$ is defined and equals $\int (f + g) d μ$ .

Facts & Assumptions

Given: The statement above.

[L1]

The actual linearity theorem only applies on $L^{1} (μ)$ (The Lebesgue integral is linear on $L^{1} (μ)$ ).

Refutation

technique · direct

1.1givenconstruct

On $R$ with Lebesgue measure, let $f : = χ_{[0, \infty)}$ ; its nonnegative Lebesgue integral is $+ \infty$ .

2.1step 1.1L1algebra∎

The integrals of $f$ , $- f$ , and $f - f = 0$ are individually defined, with [step 1.1, L1, algebra] values $+ \infty$ , $- \infty$ , and $0$ . But $\int f d λ + \int (- f) d λ = + \infty + (- \infty)$ is undefined, so the claimed unrestricted linearity identity fails. This is why [L1] restricts linearity to $L^{1}$ .

Depends on

The Lebesgue integral is linear on $L^1(\mu)$

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Gerald B. Folland, Real Analysis, 2nd ed., §2.3 (standard reference, not scraped)