The continuity of measure Let $m$ be the Lebesgue Measure. If $\{A_k\}_{k=1}^{\infty}$ is an ascending collection of measurable sets, then
$$m\left(\cup_{k=1}^\infty A_k\right)=\lim_{k\to\infty}m(A_k).$$
Can someone share a story as to why this is called one of the "continuity" properties of measure?

Since $\{A_k\}_{k=1}^\infty$ is an ascending family of sets we can vaguely write that
$$
\lim\limits_{k\to\infty} A_k=\bigcup\limits_{k=1}^\infty A_k \qquad(\color{red}{\text{note: this is not rigor!}})
$$
then this property can be written as
$$
m\left(\lim\limits_{k\to\infty} A_k\right)=\lim\limits_{k\to\infty}m(A_k)
$$
Which looks very similar to Heine definition of continuiuty. 
