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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. 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. ==Formal definition== Let (''X'', Σ) and (''Y'', Τ) be measurable spaces, meaning that ''X'' and ''Y'' are sets equipped with respective sigma algebras Σ and Τ. A function ''f'': ''X'' → ''Y'' is said to be measurable if the preimage of ''E'' under ''f'' is in Σ for every ''E'' ∈ Τ; i.e. : The notion of measurability depends on the sigma algebras Σ and Τ. To emphasize this dependency, if ''f'': ''X'' → ''Y'' is a measurable function, we will write : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Measurable function」の詳細全文を読む スポンサード リンク
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