Can someone explain me in simple words what $\sigma$-additivity is? For example, if we have a probability space ($\Omega$,$\mathcal{F}$,P), what it means that the application $ B \mapsto P(A\cap B) $ is $\sigma$-additive over $\mathcal{F}$?
I found it for the first time in probability theory and I don't understand the implication that it has in the theory.