Does additivity of probability imply almost-sure pairwise disjointedness?

Question

Suppose we have a probability measure $P$ and a countable collection of sets $\{ E_k\}_{k=1}^{\infty}$ from the corresponding $\sigma$-algebra.

If $$P\left( \bigcup_{k=1}^{\infty} E_k \right) = \sum_{k=1}^{\infty} P(E_k),$$ does that imply that $\{ E_k\}_{k=1}^n$ are almost-surely pairwise disjoint?

Answer 1

Set $F_1 = E_1$ and $F_j = E_j \setminus \bigcup_{i=1}^{j-1} E_i$ for $j \geq 2$. Then $$0= \sum_{j=1}^\infty P(E_j)-P(F_j)$$, implying $P(E_j) = P(F_j)$ for $j\geq 1$. We conclude $P(E_i \cap E_j) = 0$ if $i\neq j$.