Additive probability measures

Question

Suppose that $B$ is an algebra of sets and $p$ is a finitely additive probability measure on $B$. Show that there cannot exist uncountably many disjoint sets in the family $[A \in B: \mu (A)>0]$.

I am struggling to show this result.

Thanks

Answer 1

Hint: Let $\mathcal{P}$ be an uncountable family of disjoint sets with positive measure. For every $n$, let $\mathcal{P}_n$ be the family of those sets in $\mathcal{P}$ with measure $> \frac{1}{n}$. Can all of the $\mathcal{P}_n$ be finite? What can we conclude?