prove that $m(\emptyset) = 0$ where $m$ is countably additive over disjoint collection of sets of $\mathscr{A}$

Question

I want to prove that if $A \in \mathscr{A}$, where $\mathscr{A}$ is a $\sigma-$algebra for which $m(A) < \infty$, then $m(\emptyset) = 0$.

This is my approach.

Notice that $A \cup \emptyset = A$ and $A \cap \emptyset = \emptyset$. Then using the countably additive property, I have: $$m(A \cup \emptyset = A) = m(A) \Rightarrow m(A) + m(\emptyset) = m(A) \Rightarrow m(\emptyset) = 0$$ Is this approach correct.

Answer 1

Writing this so that this question can be marked answered.

Your approach is correct, because $(A, \emptyset)$ is a disjoint partition of $A$ and you can apply countable additivity as you did.