Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $(A_n)_n$ a sequence in $\mathcal{A}$.
If $\mathbb{P}(\bigcup_n A_n) = \sum\limits_{n} \mathbb{P}(A_n)$, then $\mathbb{P}(A_i \cap A_j) = 0$ for all $i,j \in \mathbb{N}, i \not= j$.
I have to prove that the $\sigma-$additivity of the probability measure implies that the events are pairwise disjoint.
It is clear that I could construct a disjoint sequence $B_n := A_n \setminus \bigcup_{k = 1}^{n-1} A_k$ that would satisfy the condition but that doesn't really help with this excercise.