Let $ \mathscr{F} $ be a $ \sigma $-field on a non-empty set $ \Omega $ and let $ \mu $ be a measure on $ (\Omega, \mathscr{F}) $. Prove the following properties of $ \mathscr{F} $ and $ \mu $: For disjoint $ A_{1}, \ldots, A_{m} \in \mathscr{F} $ with $ m \in \mathbb{N}, \mu\left(\bigcup_{i=1}^{m} A_{i}\right)=\sum \limits_{i=1}^{m} \mu\left(A_{i}\right) $
The definition of measure $\mu: \mathscr{F} \rightarrow [0, \infty] $ is:
(i) $\mu(\emptyset)=0$
(ii) For $ A_{n} \in \mathscr{F}, n \in \mathbb{N} $, such that $ A_{i} \cap A_{j}=\varnothing $ for all $ i, j \in \mathbb{N} $ with $ i \neq j $, $ \mu\left(\bigcup_{n=1}^{\infty} A_{n}\right)=\sum \limits_{n=1}^{\infty} \mu\left(A_{n}\right) . $
So can I just say that the property holds by definition here?
Hint: Let $B_n = A_m$ for all $1 \leq n \leq m$ and $B_n = \varnothing$ for all $n \geq m+1$. Then $(B_n)$ is a countable sequence of disjoint sets in your sigma algebra, so what does countable additivity tell you?