View source for Non-atomic measure

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" />.



====Comments====
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]].

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]].

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.

====References====
<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>