Borel measureBorelMeasure2007-10-01 17:33:332008-09-16 19:28:35DefinitionmeasureBorel measureBorel spaceBorel sigma algebra% this is the default PlanetMath preamble. as your knowledge
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{\bf Definition 1 -} Let $X$ be a topological space and $\mathcal{B}$ be its \PMlinkname{Borel $\sigma$-algebra}{BorelSigmaAlgebra}. A {\bf Borel measure} on $X$ is a measure on the measurable space $(X,\mathcal{B})$.
In the literature one can find other different definitions of Borel measure, like the following:
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\textbf{Definition 2 -} Let $X$ be a topological space and $\mathcal{B}$ be its Borel $\sigma$-algebra. A {\bf Borel measure} on $X$ is a measure $\mu$ on the measurable space $(X,\mathcal{B})$ such that $\mu (K) < \infty$ for all compact subsets $K \subset X$. (ref.\cite{MRB2k6}).
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\textbf{Definition 3 -} Let $X$ be a topological space and $\mathcal{B}$ be the $\sigma$-algebra generated by all compact sets of $X$. A {\bf Borel measure} on $X$ is a measure $\mu$ on the measurable space $(X,\mathcal{B})$ such that $\mu (K) < \infty$ for all compact subsets $K \subset X$.
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{\bf Definition 4 -} The \PMlinkname{restriction}{RestrictionOfAFunction} of the Lebesgue measure to the Borel $\sigma$-algebra of $\mathbb{R}^n$ is also sometimes called ``the'' Borel measure of $\mathbb{R}^n$.
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{\bf Remark -} Definitions $2$ and $3$ are technically different. For example, when constructing a Haar measure on a locally compact group one considers the $\sigma$-algebra generated by all compact subsets, instead of all closed (or open) sets.
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\bibitem{MRB2k6}
M.R. Buneci. 2006.,
\PMlinkexternal{Groupoid C*-Algebras.}{http://www.utgjiu.ro/math/mbuneci/preprint/p0024.pdf},
{\em Surveys in Mathematics and its Applications}, Volume 1: 71--98.
\bibitem{AC79}
A. Connes.1979. Sur la th\'eorie noncommutative de l' integration, {\em Lecture Notes in
Math.}, Springer-Verlag, Berlin, {\bf 725}: 19-14.
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