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Baire set
In mathematics, more specifically in measure theory, the Baire sets of a locally compact Hausdorff space form a σ-algebra related to the continuous functions on the space. There are several inequivalent definitions of Baire set, which all coincide for the case of locally compact σ-compact Hausdorff spaces.
The Baire sets form a subclass of the Borel sets. The converse holds in many, but not all, topological spaces.
Baire sets were introduced by , and , who named them after Baire functions that are in turn named after René-Louis Baire. They introduced them to 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.
==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.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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