= Countability of point atoms of a finite measure
{title2=$\#\{x:\mu(\{x\})\geq1/m\}\leq m\mu(\Omega)$}
For a finite <measure> on a <measurable space> whose singletons are <measurable sets>, the <set> of <point atoms of a measure> is at most <countable>. For each positive integer $m$, let $D_m$ consist of the points with singleton <measure> at least $1/m$. If $D_m$ contained more than $m\mu(\Omega)$ distinct points, <finite additivity> on those disjoint singletons would exceed the total <measure>. Thus each $D_m$ is finite.
Every strictly positive real number exceeds or equals $1/m$ for some $m$, so the <point atoms of a measure> form the <countable union> $\bigcup_{m\geq1}D_m$. This proves that the <set> is <countable>. It also shows that a <cumulative distribution function> has only countably many jumps. The result does not say the <measure> is concentrated on those points: a <non-atomic measure> may assign them total <measure> zero.
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