PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passverified 2026-09-08 (Codex)

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AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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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, where this function equals $f$ on $E$ and $0$ off $E$ (in particular, $0 \cdot (+ \infty) = 0$ here).
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).

[L2]

A sigma-algebra contains $X$ and $\emptyset$ and is closed under complements and countable unions; countable intersections follow by taking complements (Sigma-algebras).

[L3]

The rationals are countable and dense in the reals ( $Q$ is countably infinite, Both $Q$ and $R ∖ Q$ are dense in $R$ , and every nonempty open subset of $R$ is uncountable).

Proof

technique · direct

1.1L1L2L3

For any extended-real-valued $h$ , the identities ${h < a} = ⋃_{q \in Q, q < a} (X ∖ {h > q})$ and ${h > a} = ⋃_{q \in Q, q > a} (X ∖ {h < q})$ follow from rational density, including when $h (x)$ is infinite. Thus [L2] and [L3] show that measurability of all strict sublevels is equivalent to measurability of all strict superlevels.

1.2givenL1L2

Put $h = f χ_{E}$ as defined in clause 4. For $a < 0$ , ${h > a} = X$ , since $h \geq 0$ . For $a \geq 0$ , ${h > a} = E \cap {f > a}$ . Both sets are measurable, including at $a = 0$ , so clause 4 follows.

1.3L1L2L3given

For a pointwise-defined sum, ${f + g > a} = ⋃_{q \in Q} ({f > q} \cap {g > a - q})$ . Indeed, if both summands are finite and their sum exceeds $a$ , choose a rational strictly between $a - g (x)$ and $f (x)$ . If one summand is $+ \infty$ , the other is not $- \infty$ , and a rational meeting the two inequalities still exists; if a summand is $- \infty$ , the defined sum cannot exceed $a$ . The reverse inclusion follows by adding the inequalities. The union is countable and measurable, proving clause 1.

2.1L1L2step 1.1

For $c > 0$ , ${c f > a} = {f > a / c}$ ; for $c < 0$ , ${c f > a} = {f < a / c}$ . If $c = 0$ , each superlevel is either $X$ or $\emptyset$ . Step 1.1 and [L1] therefore prove clause 2.

2.2givenL1L2step 1.1

Put $u = \inf_{n} f_{n}$ and $v = \sup_{n} f_{n}$ . The defining order properties of infimum and supremum give ${u < a} = ⋃_{n} {f_{n} < a}$ and ${v > a} = ⋃_{n} {f_{n} > a}$ , including infinite values. By step 1.1 and [L2] these sets are measurable, so $u$ and $v$ are measurable by [L1]. If $f_{n} ↑ f$ , then $f = v$ pointwise. This proves clause 5.

3.1L1L2step 1.3step 2.1

The function $- f$ is measurable by step 2.1. For any real-valued measurable $h$ , the superlevel of $\max (h, 0)$ is $X$ when $a < 0$ and ${h > a}$ when $a \geq 0$ . Apply this to $h = f$ and $h = - f$ to obtain measurable $f^{+}$ and $f^{-}$ . They are finite-valued, so step 1.3 makes $∣ f ∣ = f^{+} + f^{-}$ measurable, proving clause 3.

4.1step 1.3step 2.1step 3.1step 1.2step 2.2∎

Clauses 1–5 follow respectively from steps 1.3, 2.1, 3.1, 1.2, and 2.2.

Depends on

Used by

Dependency tree · two levels

32 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, Chapter 3, Propositions 3.5–3.7 and Theorem 3.8 (standard reference, not scraped)