Continuity from above

For decreasing measurable sets, the measure of the intersection is the limit of the measures under a finiteness hypothesis.

Continuity from above

Continuity from above: Let $(X,\Sigma,\mu)$ be a measure space and let $E_1,E_2,\dots \in \Sigma$ satisfy

E_1 \supseteq E_2 \supseteq E_3 \supseteq \cdots \quad\text{and}\quad \mu(E_1)<\infty.

Then

\mu\!\left(\bigcap_{n=1}^\infty E_n\right)=\lim_{n\to\infty}\mu(E_n).

This complements continuity from below and is a basic limiting property of a measure that is used repeatedly in measure-theoretic arguments.