Suppose that a finite additive measure $\displaystyle \mu : \mathbb {M} \rightarrow [0, \infty ] $ is continuous from below, I need to show that it is a measure.

Proof.

Let $\displaystyle \{ E_n \} ^ \infty _{n=1} \subset \mathbb {M}$, all disjoint.

Define $\displaystyle F_n= \bigcup ^n _{j=1}E_j$.

Note that: 1. $\displaystyle \bigcup _{j=1} ^ \infty F_j = \bigcup _{j=1}^ \infty E_j $ and 2. $\displaystyle F_j \subset F_{j+1} $

Then we have $\displaystyle \mu ( \bigcup _{j=1}^ \infty F_j) = \lim _{n \rightarrow \infty } \mu (F_n) $

Now, $\displaystyle \mu ( \bigcup _{j=1}^ \infty E_j ) = \mu ( \bigcup _{j=1}^ \infty F_j ) = \lim _{n \rightarrow \infty } \mu (F_n) = \lim _{ n \rightarrow \infty } \mu ( \bigcup _{j=1}^n E_j )$$\displaystyle = \lim _{n \rightarrow \infty } \sum _{j=1}^n \mu (E_j) = \sum _{j=1}^ \infty \mu (E_j) $.

Is this correct? Thank you.