# Measurable function

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.

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 Zermeloâ€“Fraenkel set theory without the axiom of choice does not prove the existence of such functions.