Continuity from below is defined as
If $\{E_j\}_1^\infty \subset \mathcal{M}$ and $E_1 \subset E_2 \subset > ...$ , then $\mu(\bigcup_1^\infty E_j) = \lim_{j\rightarrow \infty} > \mu(E_j)$
There is a corresonding definition for continuity from above.
I am just starting on measure theory and is having a hard time figuring out why we have the above property. What is the point of having this property in a measure space? In probability measure space, why is it important? I am particularly looking for an intuitive reasoning for having this property.