Measurable function

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In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration. Specifically, a function between measurable spaces is said to be measurable if the preimage of each measurable set is measurable, analogous to the situation of continuous functions between topological spaces.

This definition can be deceptively simple, however, as special care must be taken regarding the σ-algebras involved. In particular, when a function f : \mathbb{R} \rightarrow \mathbb{R} is said to be Lebesgue measurable what is actually meant is that f : (\mathbb{R}, \mathcal{L}) \rightarrow (\mathbb{R}, \mathcal{B}) is a measurable function -- that is, the domain and range represent different σ-algebras on the same underlying set (here \mathcal{L} is the sigma algebra of Lebesgue measurable sets, and \mathcal{B} is the Borel algebra on \mathbb{R}). As a result, the composition of Lebesgue-measurable functions need not be Lebesgue-measurable.

By convention a topological space is assumed to be equipped with the Borel algebra generated by its open subsets unless otherwise specified. Most commonly this space will be the real or complex numbers. For instance, a real-valued measurable function is a function for which the preimage of each Borel set is measurable. A complex-valued measurable function is defined analogously. In practice, some authors use measurable functions to refer only to real-valued measurable functions with respect to the Borel algebra.[1] If the values of the function lie in an infinite-dimensional vector space instead of R or C, usually other definitions of measurability are used, such as weak measurability and Bochner measurability.

In probability theory, the sigma algebra often represents the set of available information, and a function (in this context a random variable) is measurable if and only if it represents an outcome that is knowable based on the available information. In contrast, functions that are not Lebesgue measurable are generally considered pathological, at least in the field of analysis.


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