In the literature one can find other different definitions of Borel measure, like the following:
Definition 3 - Let be a topological space and be the -algebra generated by all compact sets of . A Borel measure on is a measure on the measurable space such that for all compact subsets .
Definition 4 - The restriction (http://planetmath.org/RestrictionOfAFunction) of the Lebesgue measure to the Borel -algebra of is also sometimes called “the” Borel measure of .
Remark - Definitions and are technically different. For example, when constructing a Haar measure on a locally compact group one considers the -algebra generated by all compact subsets, instead of all closed (or open) sets.
- 1 M.R. Buneci. 2006., http://www.utgjiu.ro/math/mbuneci/preprint/p0024.pdfGroupoid C*-Algebras., Surveys in Mathematics and its Applications, Volume 1: 71–98.
- 2 A. Connes.1979. Sur la théorie noncommutative de l’ integration, Lecture Notes in Math., Springer-Verlag, Berlin, 725: 19-14.
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