measurable function

Let (X,(X)) and (Y,(Y)) be two measurable spacesMathworldPlanetmathPlanetmath. Then a function f:XY is called a measurable functionMathworldPlanetmath if:


where f-1((Y))={f-1(E)E(Y)}.

In other words, the inverse image of every (Y)-measurable set is (X)-measurable. The space of all measurable functions f:XY is denoted as


Any measurable function into (,()), where () is the Borel sigma algebra of the real numbers , is called a Borel measurable function.11More generally, a measurable function is called Borel measurable if the range space Y is a topological spaceMathworldPlanetmath with (Y) the sigma algebra generated by all open sets of Y. The space of all Borel measurable functions from a measurable space (X,(X)) is denoted by 0(X,(X)).

Similarly, we write ¯0(X,(X)) for ((X,(X)),(¯,(¯))), where (¯) is the Borel sigma algebra of ¯, the set of extended real numbers.

Remark. If f:XY and g:YZ are measurable functions, then so is gf:XZ, for if E is (Z)-measurable, then g-1(E) is (Y)-measurable, and f-1(g-1(E)) is (X)-measurable. But f-1(g-1(E))=(gf)-1(E), which implies that gf is a measurable function.


  • Let E be a subset of a measurable space X. Then the characteristic functionMathworldPlanetmathPlanetmathPlanetmathPlanetmath χE is a measurable function if and only if E is measurable.

Title measurable function
Canonical name MeasurableFunction
Date of creation 2013-03-22 12:50:50
Last modified on 2013-03-22 12:50:50
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 18
Author CWoo (3771)
Entry type Definition
Classification msc 28A20
Synonym Borel measurable
Related topic ExampleOfFunctionNotLebesgueMeasurableWithMeasurableLevelSets
Related topic LusinsTheorem2
Related topic BorelGroupoid
Related topic BorelMorphism
Defines Borel measurable function