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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Closure properties of measurable functions used by the integral

Statement

Let $(X, A, μ)$ be a measure space.

If $f, g : X \to \overline{R}$ are measurable and $f + g$ is defined pointwise, then $f + g$ is measurable.
If $c \in R$ and $f : X \to R$ is measurable, then $c f$ is measurable.
If $f : X \to R$ is measurable, then $f^{+} = \max {f, 0}$ , $f^{-} = \max {- f, 0}$ , and $∣ f ∣ = f^{+} + f^{-}$ are measurable.
If $E \in A$ and $f : X \to [0, + \infty]$ is measurable, then $f χ_{E}$ is measurable.
If $(f_{n})$ is a sequence of measurable functions $X \to \overline{R}$ , then $\inf_{n} f_{n}$ is measurable; if moreover $f_{n} ↑ f$ pointwise, then $f$ is measurable.

Facts & Assumptions

Given: A measure space $(X, A, μ)$ and functions or sets as in the relevant clause.

[L1]

A function $h : X \to \overline{R}$ is measurable exactly when ${h > a} \in A$ for every real $a$ (Extended-real-valued measurable functions).

Proof

technique · direct

1.1

If $f + g$ is defined pointwise, then for every real $a$ , [L1, algebra] ${f + g > a} = ⋃_{q \in Q} ({f > q} \cap {g > a - q}),$ so clause 1 follows from [L1].

1.2

For $c \neq 0$ one has ${c f > a} = {f > a / c}$ when $c > 0$ and [L1, algebra] ${c f > a} = {f < a / c}$ when $c < 0$ ; for $c = 0$ the function is constant. Applying [L1] proves clause 2. The formulas $f^{+} = \max {f, 0}$ and $f^{-} = \max {- f, 0}$ therefore give clause 3.

2.1

If $E \in A$ and $f \geq 0$ , then [L1, algebra] ∎ ${f χ_{E} > a} = E \cap {f > a} (a > 0),$ so clause 4 follows from [L1]. Also ${\inf_{n} f_{n} > a} = ⋂_{n} {f_{n} > a}, {\sup_{n} f_{n} > a} = ⋃_{n} {f_{n} > a},$ so clause 5 follows from [L1] as well, including the monotone-limit case $f = \sup_{n} f_{n}$ .

Depends on

Extended-real-valued measurable functions

Used by

Additivity of the nonnegative Lebesgue integral Corollary
Beppo Levi's theorem for nonnegative series Corollary
Reverse Fatou's lemma under an integrable majorant Corollary
Integrable real and complex functions, and their integrals Definition
Integral over a measurable subset Definition
Every nonnegative measurable function is the increasing limit of simple measurable functions Theorem
Fatou's lemma Theorem
Integrating against a density agrees with integrating the product Theorem
The Lebesgue integral is linear on L¹(μ) Theorem

Dependency tree · two levels

4 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 6.3 (standard reference, not scraped)
John K. Hunter, Measure Theory Notes, §4.2 (standard reference, not scraped)