In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in direct analogy to the definition that a continuous function between topological spaces preserves the topological structure: the preimage of any open set is open. In real analysis, measurable functions are used in the definition of the Lebesgue integral. In probability theory, a measurable function on a probability space is known as a random variable.
Let and be measurable spaces, meaning that and are sets equipped with respective -algebras and A function is said to be measurable if for every the pre-image of under is in ; that is, for all
That is, where is the σ-algebra generated by f. If is a measurable function, we will write
The choice of -algebras in the definition above is sometimes implicit and left up to the context. For example, for or other topological spaces, the Borel algebra (generated by all the open sets) is a common choice. Some authors define measurable functions as exclusively real-valued ones with respect to the Borel algebra.
If the values of the function lie in an infinite-dimensional vector space, other non-equivalent definitions of measurability, such as weak measurability and Bochner measurability, exist.
Real-valued functions encountered in applications tend to be measurable; however, it is not difficult to prove the existence of non-measurable functions. Such proofs rely on the axiom of choice in an essential way, in the sense that Zermelo–Fraenkel set theory without the axiom of choice does not prove the existence of such functions.
In any measure space with a non-measurable set one can construct a non-measurable indicator function:
As another example, any non-constant function is non-measurable with respect to the trivial -algebra since the preimage of any point in the range is some proper, nonempty subset of which is not an element of the trivial