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'Borel measure'
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| Title of object: |
Borel measure |
| Canonical Name: |
BorelMeasure |
| Type: |
Definition |
| Created on: |
2007-10-01 17:33:33 |
| Modified on: |
2008-09-15 13:15:56 |
| Classification: |
msc:28A10, msc:28A12, msc:28C15, msc:60A10 |
| Keywords: |
Borel measure, Borel space, Borel sigma algebra |
Preamble:
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Content:
{\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 $(X,\mathcal{B})$.
An alternative definition of a Borel measure is as follows.
\textbf{Definition 0.1}
Let $X$ be a topological space and let $\mu$ be a \PMlinkname{measure}{Measure} on the
\PMlinkname{measurable space}{MeasurableSpace} $(X,\mathcal{B})$.
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} of $X$, (ref.\cite{MRB2k6}).
{\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 $\mathbb{R}^n$.
\begin{thebibliography}{9}
\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.
\end{thebibliography} |
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