Measurable function

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. A function f : S -> R (the real numbers) is called measurable if it is measurable with respect to the Borel σ-algebra on R.

Only measurable functions can be integrated. All random variables are 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.