Measurable function

In mathematics, measurable functionswell-behaved functions between measurable spaces. Almost all functions studiedanalysismeasurable.

If X isσ-algebra over SY isσ-algebra over T, thenfunction f : S -> Tcalled measurable ifpreimageevery setYin X.

By convention, if Tsome topological space, such asreal numbers R orcomplex numbers C, then the Borel σ-algebra on Tused, unless otherwise specified.

The compositiontwo measurable functionsmeasurable.

Only measurable functions can be integrated. Random variablesby definition measurable functions defined on probability spaces.

Any continuous function from one topological spaceanothermeasurablerespect toBorel σ-algebras ontwo spaces.

See also: σ-algebra