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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.

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A map $m\colon \mathcal F\to \Bbb R_{\geq 0}$ is $\sigma$-additive if for each sequence $\{B_n\}\subset \mathcal F$ of pairwise disjoint measurable sets, we have $$m\left(\bigcup_{n=1}^{+\infty}B_n\right)=\sum_{n=1}^{+\infty}m(B_n).$$ This is the formal definition.

This means that the "measure" of a disjoint union is the sum of the "measures" of each elements of the union. The letter $\sigma$ expresses an idea of countability, as we can work with maps finitely-additive. It allows us to work with sequences of sets.

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An $f$ function is additive, if for any finite sequence of elements $a_i$, $$f\left(\sum_i a_i\right) = \sum_i f(a_i)$$ Then, $f$ is $\sigma$-additive, if it also holds for any countable infinite set $\{a_i\}_i$ of elements.

In measure (and probabilistic) context, it is only said so for pairwise disjoint sets in place of the $a_i$'s, and $\sum a_i$ is meant their union. And its meaning is the usual 'additivity' property of the area (and volume and similar) notion.

$P$ is a probability measure, so, by definition it is assumed to be $\sigma$-additive, and so is $B\mapsto P(B\cap A)$.

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