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In mathematics, Baire functions are functions obtained from continuous functions by transfinite iteration of the operation of forming pointwise limits of sequences of functions. They were introduced by René-Louis Baire in 1899. A Baire set is a set whose characteristic function is a Baire function. ==Classification of Baire functions== Baire functions of class α, for any countable ordinal number α, form a vector space of real-valued functions defined on a topological space, as follows. *The Baire class 0 functions are the continuous functions. *The Baire class 1 functions are those functions which are the pointwise limit of a sequence of Baire class 0 functions. *In general, the Baire class α functions are all functions which are the pointwise limit of a sequence of functions of Baire class less than α. Some authors define the classes slightly differently, by removing all functions of class less than α from the functions of class α. This means that each Baire function has a well defined class, but the functions of given class no longer form a vector space. Henri Lebesgue proved that (for functions on the unit interval) each Baire class of a countable ordinal number contains functions not in any smaller class, and that there exist functions which are not in any Baire class. 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「Baire function」の詳細全文を読む スポンサード リンク
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