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.
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.
If a function from one topological space to another is measurable with respect to the Borel σ-algebras on the two spaces, the function is also known as a Borel function.
Continuous functions are Borel, however, not all Borel functions are continuous.
See also: σ-algebra