Baire function


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 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 that is not of class 1 is the characteristic function of the rational numbers,, also known as the Dirichlet function which is discontinuous everywhere.

Baire class 3

An example of such functions is given by the indicator of the set of normal numbers, which is a Borel set of rank 3.