Continuity from below and above

In Folland's Real analysis, two of properties of measures are stated as follows:

Let $(X,\mathcal{M}, \mu)$ be a measure space.

Continuity from below: If $\{E_j\}_1^{\infty} \subset \mathcal{M}$ and $E_1 \subset E_2 \subset \cdots$, then $\mu(\bigcup_1^{\infty} E_j) = \lim_{j \to \infty} \mu(E_j)$

Similarly,

Continuity from above: If $\{E_j\}_1^{\infty} \subset \mathcal{M}$, $E_1 \supset E_2 \supset \cdots$ and $\mu(E_1) < \infty$. Then $\mu(\bigcap_1^{\infty} E_j) = \lim_{j \to \infty} \mu(E_j)$

I do not understand the point of these statements. How they are related to continuity for example?

Solution 1:

It's reasonable for a continuous increasing function $f$ with real values, defined on the real line, to have $f(t_n)\uparrow f(t)$ when $t_n\uparrow t$ and $f(t_n)\downarrow f(t)$ when $t_n\downarrow t$. As a measure can take possibly infinite values, we have to be careful in this context, but that's the idea.

The set functions $\mu\colon\cal M\to \overline{\Bbb R}_+$ is increasing in the sense that if $A\subset B$ then $\mu(A)\subset \mu(B)$. If $A_n\uparrow A$, then $\mu(A_n)\uparrow \mu(A)$.

We have to assumed that at least a set in $\{B_n\}$ is of finite measure in the case of decreasing sequence, to avoid counter-examples likes $X=\Bbb R$ with Lebesgue measure and $B_n=[n,+\infty)$ (like if we had a function defined on the real line with values in $\overline{\Bbb R}_+$ and such that $f(x)=0$ if $x\leqslant 0$, $f(x)=+\infty$ otherwise).