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| − | A [[Measure|measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066950/n0669501.png" /> on a measurable space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066950/n0669502.png" /> for which there are no atoms of positive measure, i.e. sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066950/n0669503.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066950/n0669504.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066950/n0669505.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066950/n0669506.png" /> imply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066950/n0669507.png" />.
| + | #REDIRECT[[Atom]] |
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| − | ====Comments====
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| − | An atom in a measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066950/n0669508.png" /> is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066950/n0669509.png" /> for which i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066950/n06695010.png" />; and ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066950/n06695011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066950/n06695012.png" /> imply either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066950/n06695013.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066950/n06695014.png" />. See also [[Atom|Atom]].
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| − | A measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066950/n06695015.png" /> is called non-atomic if no element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066950/n06695016.png" /> is an atom. In probability theory measure spaces build up completely from atoms, i.e. using atomic measures, frequently occur, cf. [[Atomic distribution|Atomic distribution]].
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| − | A probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066950/n06695017.png" /> decomposes as a sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066950/n06695018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066950/n06695019.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066950/n06695020.png" /> is an atomic distribution and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066950/n06695021.png" /> a [[Continuous distribution|continuous distribution]], i.e. a non-atomic one. This goes by the name Jordan decomposition theorem.
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| − | ====References====
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| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Feller, [[Feller, "An introduction to probability theory and its applications"|"An introduction to probability theory and its applications"]], '''2''' , Wiley (1971) pp. 135</TD></TR></table>
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Latest revision as of 10:36, 17 September 2012
How to Cite This Entry:
Non-atomic measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Non-atomic_measure&oldid=27993
This article was adapted from an original article by N.N. Vorob'ev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article