Baire classes

2020 Mathematics Subject Classification: Primary: 54E52 [MSN][ZBL]

The Baire classes are families of real functions on a topological space $X$, indexed by countable ordinal numbers, which are defined inductively iterating the operation of taking pointwise limits of sequences of functions.

Definition

More precisely:

• The zero-th Baire class $\mathcal{H}_0$ is the class of continuous functions;

• The first Baire class $\mathcal{H}_1$ consists of those functions which are discontinuous but are pointwise limits of sequences of continuous functions, i.e. the minimal class of functions containing continuous functions and closed under the operation of taking pointwise limits;

• If $\alpha$ is any countable ordinal number, the $\alpha$ Baire class $\mathcal{H}_\alpha$ consists of those functions which do not belong to any class $\mathcal{H}_\beta$ with $\beta < \alpha$ but are pointwise limits of any sequence of functions $\{f_k\}\subset \cup_{\beta<\alpha} \mathcal{H}_\beta$.

The union of all such classes form the Baire functions. They were first defined in 1899 by R. Baire (see [Ba]) when $X$ is the standard real line and are also known as the Baire classification. Although the definition makes sense for general topological space, the classes are most commonly used when $X$ is a (complete) metric space or a space with the Baire property.

Properties

Given any countable ordinal $\alpha$, the union of all Baire classes $\mathcal{H}_\beta$ with $\beta\leq \alpha$ is closed under the operations of taking linear combinations, products and quotients (for non-zero denominators) and uniform limits. Necessary and sufficient conditions have been established for a sequence of functions in a Baire class not higher than $\alpha$ to converge to a function in a Baire class not higher than $\alpha$, see [Gag].

For many topological spaces $X$ all Baire classes are nonempty. More precisely, given a complete metric space $X$ consider the kernel of $X$, i.e. the union of all subsets of $X$ which contain no (relatively) isolated points. If such kernel is not empty, then none of the Baire classes is empty (see [Ha]; this theorem was first proved by Lebesgue when $X$ is an interval of the real line).

When $X$ is the standard Euclidean space $\mathbb R^n$, the Baire functions are all Borel measurable and hence Lebesgue measurable. A partial converse of this fact states that any Lebesgue-measurable function coincides, up to a set of measure zero, with a Baire function of class at most $2$ (see [Nat]).

Baire-1 functions

The functions in the first class are often called Baire-1 functions and arise naturally in several problems (note, for instance, that the derivative of a differentiable function is a Baire-1 function). Baire himself made the most detailed study of functions of such class (when $X=\mathbb R$). In particular he showed that a necessary and sufficient condition for a discontinuous function to belong to the first class is the existence of a point of continuity of the induced function on each perfect set (Baire's characterization theorem). Thus the Dirichlet's function, which takes the values $0$ on the irrational numbers and $1$ on the rational numbers, is a classical example of function which does not belong to the first Baire class (indeed it belongs to the second class).

The Baire's characterization theorem is applicable when the domain $X$ has the Baire property, see [Ha]. A very useful byproduct is that the points of discontinuity of a Baire-1 function is a residual set.

References

[Ba]	R. Baire, "Leçons sur les fonctions discontinues, professées au collège de France" , Gauthier-Villars (1905) Zbl 36.0438.01
[Bo]	R.P Boas jr., "A primer of real functions" , Math. Assoc. Amer. (1981) MR0652482 Zbl 0473.26002
[Gag]	B.M. Gagaeff, "Sur les suites convergentes de fonctions mesurables $\mathbf B^n$" Fund. Math. , 18 (1932) pp. 182–188 Zbl 0004.20501
[Ha]	F. Hausdorff, "Grundzüge der Mengenlehre" , Leipzig (1914) (Reprinted (incomplete) English translation: Set theory, Chelsea (1978)) MR1034865 MR0979016 MR0031025 Zbl 1175.01034 Zbl 45.0123.01
[Na]	I.P. Natanson, "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M. (1961) (Translated from Russian) MR0640867 MR0409747 MR0259033 MR0063424 Zbl 0097.26601
[Ox]	J.C. Oxtoby, "Measure and category" , Springer (1971) MR0393403 0217.09201 Zbl 0217.09201
[Ro]	H.L. Royden, "Real analysis", Macmillan (1968) MR0151555 Zbl 0197.03501
[vR]	A.C.M. van Rooy, W.H. Schikhof, "A second course on real functions" , Cambridge Univ. Press (1982) MR655599