DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-27

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A measurable function between measurable spaces

Definition

Let $(X, A)$ and $(Y, B)$ be measurable spaces (Measurable spaces and measurable sets). A function $f : X \to Y$ is measurable when

$f^{- 1} (B) \in A for every B \in B .$

Equivalently, the family

$C_{f} : = {B \subseteq Y : f^{- 1} (B) \in A}$

is a sigma-algebra on $Y$ containing $B$ .

Depends on

Measurable spaces and measurable sets

Used by

A simple function and its canonical representation Definition
Borel measurable and Lebesgue measurable functions on ℝⁿ Definition
The sigma-algebra generated by a function Definition
An indicator function is measurable exactly when its set is measurable Proposition
A generating family on the codomain suffices to test measurability Theorem
Composition with a Borel measurable outer map preserves measurability Theorem

Dependency tree · two levels

2 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

Sheldon Axler, Measure, Integration and Real Analysis, Section 2B (standard reference, not scraped)
John K. Hunter, Measure Theory, Definition 3.1 (standard reference, not scraped)