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In mathematics, measurable functions are well-behaved functions between measurable spaces.
Almost all functions studied in analysis are
measurable.
If X is a σ-algebra over S and Y is a
σ-algebra over T, then a function f : S -> T is called
measurable if the preimage of every set in Y is in X.
By convention, if T is some topological space, such
as the real numbers R or the complex numbers C, then the Borel
σ-algebra on T is used, unless otherwise specified.
The composition of two measurable functions 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.
See also: σ-algebra
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