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 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, one writes to emphasize the dependency on the -algebras and 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. Random variables are by definition measurable functions defined on probability spaces. If and are Borel spaces, a measurable function is also called a Borel function. Continuous functions are Borel functions but not all Borel functions are continuous. However, a measurable function is nearly a continuous function; see Luzin's theorem. If a Borel function happens to be a section of a map it is called a Borel section. A Lebesgue measurable function is a measurable function where is the -algebra of Lebesgue measurable sets, and is the Borel algebra on the complex numbers Lebesgue measurable functions are of interest in mathematical analysis because they can be integrated.