= Point atom of a measure
{title2=$\mu(\{x\})>0$}
= Point atoms of a measure
{synonym}
A point atom is a point $x$ whose singleton is a <measurable set> with strictly positive <measure>. That singleton is an <atom of a measure>. This is more specific than a general <atom of a measure>, which is defined as a positive-measure <set> that cannot be divided into two <measurable sets> of positive <measure>.
For a <probability measure> on the <Borel sigma-algebra> of $\mathbb R$, a point atom at $x$ appears as a jump of its <cumulative distribution function>:
$$
\mu(\{x\})=F_\mu(x)-F_\mu(x-).
$$
<Continuity of probability> applied to intervals approaching $x$ proves this identity. Keeping such jumps is essential when deciding whether an endpoint belongs to an <integration> region.
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