 Solution

The probability measure $\mathbb{P}$ maps the sigma-algebra $\mathcal{F}$ of events into $[0,1]$. Its probability axioms are nonnegativity, normalization $\mathbb{P}(\Omega )=1$, and countable additivity: for every pairwise disjoint sequence of events $A_1,A_2,\dots$,The sample space is $\Omega$, and $\mathcal{F}$ specifies which subsets are events. Requiring countable additivity, rather than only its finite version, is part of the probability space definition.