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Difference between revisions of "Measurable space"

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A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063230/m0632301.png" /> with a distinguished ring or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063230/m0632302.png" />-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063230/m0632303.png" /> (in particular, an algebra or a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063230/m0632304.png" />-algebra) of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063230/m0632305.png" />.
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$ \newcommand{\R}{\mathbb R}
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\newcommand{\A}{\mathcal A}
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\newcommand{\P}{\mathbf P} $
  
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Examples: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063230/m0632306.png" /> with the ring of Jordan-measurable sets (see [[Jordan measure|Jordan measure]]); <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063230/m0632307.png" /> with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063230/m0632308.png" />-ring of sets of finite [[Lebesgue measure|Lebesgue measure]]; a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063230/m0632309.png" /> with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063230/m06323010.png" />-algebra of Borel sets (cf. [[Borel set|Borel set]]).
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A set with a distinguished σ-algebra of subsets (called measurable). More formally: a pair $(X,\A)$ consisting of a set $X$ and a σ-algebra $\A$ of subsets of $X$.
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Examples: $\R^n$ with the Borel σ-algebra; $\R^n$ with the Lebesgue σ-algebra.
  
 
====References====
 
====References====
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> P.R. Halmos,   "Measure theory" , v. Nostrand  (1950)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">Terence Tao, "An introduction to measure theory", AMS (2011)</TD></TR></table>

Revision as of 17:59, 20 December 2011

$ \newcommand{\R}{\mathbb R} \newcommand{\Om}{\Omega} \newcommand{\A}{\mathcal A} \newcommand{\P}{\mathbf P} $

A set with a distinguished σ-algebra of subsets (called measurable). More formally: a pair $(X,\A)$ consisting of a set $X$ and a σ-algebra $\A$ of subsets of $X$.

Examples: $\R^n$ with the Borel σ-algebra; $\R^n$ with the Lebesgue σ-algebra.

References

[1] Terence Tao, "An introduction to measure theory", AMS (2011)
How to Cite This Entry:
Measurable space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Measurable_space&oldid=19851
This article was adapted from an original article by V.V. Sazonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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