When is the algebra of regular sets for a finitely additive borel measure a α-algebra?

Journal of the Australian Mathematical Society

Thomas E. Armstrong¹ 

关键词: 28 C 15;    28 A 60;    54 G 10;   

DOI  :  10.1017/S1446788700018802

学科分类：数学（综合）

来源: Cambridge University Press

【 摘 要 】

It is shown that hte algebra of regular sets for a finitely additive Borel measure μ on a compact Hausdroff space is a σ-algebra only if it includes the Baire algebra and μ is countably additive onthe σ-algebra of regular sets. Any infinite compact Hausdroff space admits a finitely additive Borel measure whose algebra of regular sets is not a σ-algebra. Although a finitely additive measure with a σ-algebra of regular sets is countably additive on the Baire σ-algebra there are examples of finitely additive extensions of countably additive Baire measures whose regular algebra is not a σ-algebra. We examine the particular case of extensions of Dirac measures. In this context it is shown that all extensions of a {0, 1}-valued countably additive measure from a σ-algebra to a larger σ-algebra are countably additive if and only if the convex set of these extensions is a finite dimensional simplex.

【 授权许可】

Unknown