In mathematics, particularly in measure theory, measurable functions are structurepreserving 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 σalgebras involved. In particular, when a function is said to be Lebesgue measurable what is actually meant is that is a measurable function  that is, the domain and range represent different σalgebras on the same underlying set (here is the sigma algebra of Lebesgue measurable sets, and is the Borel algebra on ). As a result, the composition of Lebesguemeasurable functions need not be Lebesguemeasurable.
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 realvalued measurable function is a function for which the preimage of each Borel set is measurable. A complexvalued measurable function is defined analogously. In practice, some authors use measurable functions to refer only to realvalued measurable functions with respect to the Borel algebra.^{[1]} If the values of the function lie in an infinitedimensional 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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