Borel measure

Definition 1 - Let $X$ be a topological space and $ℬ$ be its Borel $\sigma$-algebra (http://planetmath.org/BorelSigmaAlgebra). A Borel measure on $X$ is a measure on the measurable space $(X,ℬ)$.

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 $ℬ$ be its Borel $\sigma$-algebra. A Borel measure on $X$ is a measure $\mu$ on the measurable space $(X,ℬ)$ such that for all compact subsets $K\subset X$. (ref.[1]).

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Definition 3 - Let $X$ be a topological space and $ℬ$ be the $\sigma$-algebra generated by all compact sets of $X$. A Borel measure on $X$ is a measure $\mu$ on the measurable space $(X,ℬ)$ such that for all compact subsets $K\subset X$.

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Definition 4 - The restriction (http://planetmath.org/RestrictionOfAFunction) of the Lebesgue measure to the Borel $\sigma$-algebra of ${ℝ}^n$ is also sometimes called “the” Borel measure of ${ℝ}^n$.

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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.