Definition 1 - Let $X$ be a topological space and $\mathcal{B}$ be its Borel $\sigma$ -algebra. A 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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Definition 2 - Let $X$ be a topological space and $\mathcal{B}$ be its Borel $\sigma$ -algebra. A 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.[1]).

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Definition 3 - Let $X$ be a topological space and $\mathcal{B}$ be the $\sigma$ -algebra generated by all compact sets of $X$ . A 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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Definition 4 - The restriction 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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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.

Bibliography

1

M.R. Buneci. 2006., Groupoid 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.