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Version 15 |
| {\bf \PMlinkescapetext{Definition} -} 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 $(X,\mathcal{B})$. |
{\bf \PMlinkescapetext{Definition} -} 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 $(X,\mathcal{B})$. |
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| An alternative definition of a Borel measure is as follows. |
An alternative definition of a Borel measure is as follows. |
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| \textbf{Definition 0.1} |
\textbf{Definition 0.1} |
| Let $X$ be a topological space and let $\mu$ be a \PMlinkname{measure}{Measure} on the |
Let $X$ be a topological space and let $\mu$ be a \PMlinkname{measure}{Measure} on the |
| \PMlinkname{measurable space}{MeasurableSpace} $(X,\mathcal{B})$, with $\mathcal{B}$ being |
\PMlinkname{measurable space}{MeasurableSpace} $(X,\mathcal{B})$, with $\mathcal{B}$ being |
| the $\sigma$-algebra of the Borel sets of $X$. Then, a \emph{Borel measure on $X$} is defined as a |
the $\sigma$-algebra of the Borel sets of $X$. Then, a \emph{Borel measure on $X$} is defined as a |
| \emph{measure $\mu_B$ with the property that $\mu_B (K) < \infty$ for all compact subsets $K$} of $X$, (ref.\cite{MRB2k6}). |
\emph{measure $\mu_B$ with the property that $\mu_B (K) < \infty$ for all compact subsets $K$} of $X$, (ref.\cite{MRB2k6}). |
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| {\bf Remark -} 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 $\mathbb{R}^n$. |
{\bf Remark -} 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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| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
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| \bibitem{MRB2k6} |
\bibitem{MRB2k6} |
| M.R. Buneci. 2006., |
M.R. Buneci. 2006., |
| \PMlinkexternal{Groupoid C*-Algebras.}{http://www.utgjiu.ro/math/mbuneci/preprint/p0024.pdf}, |
\PMlinkexternal{Groupoid C*-Algebras.}{http://www.utgjiu.ro/math/mbuneci/preprint/p0024.pdf}, |
| {\em Surveys in Mathematics and its Applications}, Volume 1: 71--98. |
{\em Surveys in Mathematics and its Applications}, Volume 1: 71--98. |
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| \bibitem{AC79} |
\bibitem{AC79} |
| A. Connes.1979. Sur la th\'eorie noncommutative de l' integration, {\em Lecture Notes in |
A. Connes.1979. Sur la th\'eorie noncommutative de l' integration, {\em Lecture Notes in |
| Math.}, Springer-Verlag, Berlin, {\bf 725}: 19-14. |
Math.}, Springer-Verlag, Berlin, {\bf 725}: 19-14. |
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| \end{thebibliography} |
\end{thebibliography} |
