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. 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.

Baire class 1

Examples: The Baire Characterisation Theorem states that a real valued function f defined on a Banach space X is a Baire-1 function if and only if for every non-empty closed subset K of X, the restriction of f to K has a point of continuity relative to the topology of K. By another theorem of Baire, for every Baire-1 function the points of continuity are a comeager Gδ set.

Baire class 2

An example of a Baire class 2 function on the interval [0,1] that is not of class 1 is the characteristic function of the rational numbers,, also known as the Dirichlet function which is discontinuous everywhere.

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General