Measurable function

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.