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In mathematics, measurable functions are well-behaved functions between measurable spaces. Functions studied in analysis that are not measurable are generally considered pathological.
If X is a σ-algebra over S and Y is a σ-algebra over T, then a function f : S → T is measurable if the preimage of every set in Y is in X.
By convention, if T is some topological space, such as the space of real numbers R or the complex numbers C, then the Borel σ-algebra generated by the open sets on T is used, unless otherwise specified.
In general, the composition fg of two measurable functions is not measurable, but it is measurable
when f is continuous and g is measurable.
Only measurable functions can be integrated. Random variables are by definition measurable functions defined on probability spaces.
Any continuous function from one topological space to another is measurable with respect to the Borel σ-algebras on the two spaces. In this case measurable functions are also called Borel functions.
See also: σ-algebra
de:messbare Funktion pl:funkcja mierzalna
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