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| − | A probability distribution on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013860/a0138601.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013860/a0138602.png" /> of subsets of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013860/a0138603.png" /> concentrated on the set of its atoms (cf. [[Atom|Atom]]). A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013860/a0138604.png" /> in a probability space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013860/a0138605.png" /> is called an atom of the distribution if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013860/a0138606.png" /> and if it follows from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013860/a0138607.png" /> that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013860/a0138608.png" /> or that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013860/a0138609.png" />. In the particular case in which the atoms are discrete points, an atomic distribution coincides with a [[Discrete distribution|discrete distribution]].
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Latest revision as of 10:34, 17 September 2012
How to Cite This Entry:
Atomic distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Atomic_distribution&oldid=18055
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article