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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Topological spaces whose Baire measure admits a regular Borel extension


Authors: Haruto Ohta and Ken-ichi Tamano
Journal: Trans. Amer. Math. Soc. 317 (1990), 393-415
MSC: Primary 28C15; Secondary 54C50, 54G20
DOI: https://doi.org/10.1090/S0002-9947-1990-0946425-5
MathSciNet review: 946425
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Abstract: A completely regular, Hausdorff space $ X$ is called a Măík space if every Baire measure on $ X$ admits an extension to a regular Borel measure. We answer the questions about Măík spaces asked by Wheeler [29] and study their topological properties. In particular, we give examples of the following spaces: A locally compact, measure compact space which is not weakly Bairedominated; i.e., it has a sequence $ {F_n} \downarrow \emptyset $ of regular closed sets such that $ { \cap _{n \in \omega }}{B_n} \ne \emptyset $ whenever $ {B_n}$'s are Baire sets with $ {F_n} \subset {B_n}$; a countably paracompact, non-Măík space; a locally compact, non-Măík space $ X$ such that the absolute $ E(X)$ is a Măík space; and a locally compact, Măík space $ X$ for which $ E(X)$ is not. It is also proved that Michael's product space is not weakly Baire-dominated.


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DOI: https://doi.org/10.1090/S0002-9947-1990-0946425-5
Keywords: Măík space, countably paracompact, locally compact, absolute, Michael line, Baire measure, Borel measure, regular Borel extension, measure compact
Article copyright: © Copyright 1990 American Mathematical Society

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