Baire set

There are several inequivalent definitions of Baire sets, but in the most widely used, the Baire sets of a locally compact Hausdorff space form the smallest σ-algebra such that all compactly supported continuous functions are measurable. Thus, measures defined on this σ-algebra, called Baire measures, are a convenient framework for integration on locally compact Hausdorff spaces. In particular, any compactly supported continuous function on such a space is integrable with respect to any finite Baire measure.

Every Baire set is a Borel set. The converse holds in many, but not all, topological spaces. Baire sets avoid some pathological properties of Borel sets on spaces without a countable base for the topology. In practice, the use of Baire measures on Baire sets can often be replaced by the use of regular Borel measures on Borel sets.

Baire sets were introduced by Kunihiko Kodaira (1941, Definition 4), Shizuo Kakutani and Kunihiko Kodaira (1944) and Halmos (1950, page 220), who named them after Baire functions, which are in turn named after René-Louis Baire.

Basic definitions

There are at least three inequivalent definitions of Baire sets on locally compact Hausdorff spaces, and even more definitions for general topological spaces, though all these definitions are equivalent for locally compact σ-compact Hausdorff spaces. Moreover, some authors add restrictions on the topological space that Baire sets are defined on, and only define Baire sets on spaces that are compact Hausdorff, or locally compact Hausdorff, or σ-compact.

First definition

Kunihiko Kodaira defined what we call Baire sets (although he confusingly calls them "Borel sets") of certain topological spaces to be the sets whose characteristic function is a Baire function (the smallest class of functions containing all continuous real-valued functions and closed under pointwise limits of sequences). Dudley (1989, Sect. 7.1) gives an equivalent definition and defines Baire sets of a topological space to be elements of the smallest σ-algebra such that all continuous real-valued functions are measurable. For locally compact σ-compact Hausdorff spaces this is equivalent to the following definitions, but in general the definitions are not equivalent.

Conversely, the Baire functions are exactly the real-valued functions that are Baire measurable. For metric spaces, the Baire sets coincide with the Borel sets.

Second definition

Halmos (1950, page 220) defined Baire sets of a locally compact Hausdorff space to be the elements of the σ-ring generated by the compact G_δ sets. This definition is no longer used much, as σ-rings are somewhat out of fashion. When the space is σ-compact, this definition is equivalent to the next definition.

One reason for working with compact G_δ sets rather than closed G_δ sets is that Baire measures are then automatically regular (Halmos 1950, theorem G page 228).

Third definition

The third and most widely used definition is similar to Halmos's definition, modified so that the Baire sets form a σ-algebra rather than just a σ-ring.

A subset of a locally compact Hausdorff topological space is called a Baire set if it is a member of the smallest σ–algebra containing all compact G_δ sets. In other words, the σ–algebra of Baire sets is the σ–algebra generated by all those intersections of countably many open sets that yield a compact set. Alternatively, Baire sets form the smallest σ-algebra such that all continuous functions of compact support are measurable (at least on locally compact Hausdorff spaces; on general topological spaces these two conditions need not be equivalent).

For σ-compact spaces this is equivalent to Halmos's definition. For spaces that are not σ-compact the Baire sets under this definition are those under Halmos's definition together with their complements. However, in this case it is no longer true that a finite Baire measure is necessarily regular: for example, the Baire probability measure that assigns measure 0 to every countable subset of an uncountable discrete space and measure 1 to every co-countable subset is a Baire probability measure that is not regular.