DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-27

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Extended-real-valued measurable functions

Definition

Let $(X, A, μ)$ be a measure space (Measure spaces). Equip $\overline{R}$ (The extended real line $\overline{R} = R \cup {- \infty, + \infty}$ , its order, and the arithmetic that is left undefined) with its order topology and let $B (\overline{R})$ be the corresponding Borel sigma-algebra (The Borel sigma-algebra of a topological space).

A function $f : X \to \overline{R}$ is measurable when $f^{- 1} (B) \in A$ for every $B \in B (\overline{R})$ .

Equivalently, it is enough to require ${x \in X : f (x) > a} \in A for every a \in R,$ because the rays $(a, + \infty]$ generate $B (\overline{R})$ .

Depends on

Used by

A nonintegrable observable with divergent ergodic averages Counterexample
Extremal length and the curve-family modulus of a path family Definition
Nonnegative simple measurable functions Definition
Pointwise Borel nonnegative densities Definition
States, tracial states and faithful normal traces on a von Neumann algebra Definition
The nonnegative Lebesgue integral Definition
Birkhoff's theorem requires integrability False statement
Counting measure on a discrete group is Haar, Haar measures there are its multiples, and integrals against them are sums Lemma
Generic evaluation of bounded measurable functions by rational cuts Lemma
Closure properties of measurable functions used by the integral Proposition
Every nonnegative measurable function is the increasing limit of simple measurable functions Theorem

Dependency tree · two levels

13 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, ch. 6 (standard reference, not scraped)
Gerald B. Folland, Real Analysis, 2nd ed., §2.2 (standard reference, not scraped)