Continuity from below

For increasing measurable sets, the measure of the union is the limit of the measures.

Continuity from below

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

E_1 \subseteq E_2 \subseteq E_3 \subseteq \cdots.

Then

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

Together with continuity from above , this expresses how measures interact with countable unions and intersections of measurable sets .