 Point atom of a measure

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: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.