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In mathematics, more specifically in measure theory, the Baire sets form a σ-algebra of a topological space that avoids some of the pathological properties of Borel sets. 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.

Attributes	Values
rdfs:label	Baire set (en) ベール集合 (ja) 베르 집합 (ko)
rdfs:comment	측도론에서 베르 집합(Baire集合, 영어: Baire set)은 실수 값 연속 함수들을 모두 가측 함수로 만드는 가장 엉성한 시그마 대수이다. 구체적으로, Gδ 콤팩트 집합들로 생성된다. 베르 집합들의 시그마 대수는 보렐 시그마 대수의 부분 시그마 대수이다. (ko) 数学、特に測度論においてベール集合は測度論と位相空間論の関係の理解に重要な概念である。とくに、ベール集合の理解は距離付不能な位相空間での測度の扱いに関する直観を助ける。ベール集合はボレル集合のサブクラスである。逆も全てではないが多くの重要な位相空間で成り立つ。 (ja) In mathematics, more specifically in measure theory, the Baire sets form a σ-algebra of a topological space that avoids some of the pathological properties of Borel sets. 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. (en)
dcterms:subject	General topology Measure theory
Wikipage page ID	3976202 (xsd:integer)
Wikipage revision ID	1120601733 (xsd:integer)
Link from a Wikipage to another Wikipage	Borel sets René-Louis Baire Σ-algebra Compact space Countable set Mathematics Measure theory G-delta set Standard probability space Locally compact Measure (mathematics) Complete measure Measurable function Hausdorff space Locally compact space Euclidean space Kolmogorov extension theorem Regular measure Baire function Baire measure General topology Measure theory Support (mathematics) Borel set Kunihiko Kodaira Tychonoff's theorem Probability measure Topological space Σ-compact space Discrete topological space Sigma ring
sameAs	Baire set Baire set Baire set Baire set Baire set Baire set
dbp:wikiPageUsesTemplate	dbt:Springer dbt:Citation dbt:Cite_book dbt:For dbt:Harv dbt:Harvtxt dbt:Reflist dbt:Harvs dbt:Measure_theory
first	Kunihiko (en) Shizuo (en)
id	p/b015050 (en)
last	Kodaira (en) Kakutani (en)
title	Baire set (en)
year	1941 (xsd:integer) 1944 (xsd:integer)
loc	Definition 4 (en)
has abstract	In mathematics, more specifically in measure theory, the Baire sets form a σ-algebra of a topological space that avoids some of the pathological properties of Borel sets. 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 , Shizuo Kakutani and Kunihiko Kodaira and Halmos , who named them after Baire functions, which are in turn named after René-Louis Baire. (en) 측도론에서 베르 집합(Baire集合, 영어: Baire set)은 실수 값 연속 함수들을 모두 가측 함수로 만드는 가장 엉성한 시그마 대수이다. 구체적으로, Gδ 콤팩트 집합들로 생성된다. 베르 집합들의 시그마 대수는 보렐 시그마 대수의 부분 시그마 대수이다. (ko) 数学、特に測度論においてベール集合は測度論と位相空間論の関係の理解に重要な概念である。とくに、ベール集合の理解は距離付不能な位相空間での測度の扱いに関する直観を助ける。ベール集合はボレル集合のサブクラスである。逆も全てではないが多くの重要な位相空間で成り立つ。 (ja)
prov:wasDerivedFrom	wikipedia-en:Baire_set?oldid=1120601733&ns=0
page length (characters) of wiki page	9049 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Baire_set
is Link from a Wikipage to another Wikipage of	List of eponyms (A–K) René-Louis Baire Complete Boolean algebra Haar measure Baire function Baire measure Kunihiko Kodaira Set function List of types of sets The Banach–Tarski Paradox (book) Baire-measurable function Baire measurable function
is Wikipage redirect of	Baire-measurable function Baire measurable function
is known for of	René-Louis Baire
is known for of	René-Louis Baire
is foaf:primaryTopic of	wikipedia-en:Baire_set

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