> [!NOTE] Definition 1 (Probability Measure) > Let $\Omega$ be a [[Sample Space|sample space]]. Let $\mathcal{F}$ be an [[Event Space|event space]] on $\Omega.$ A probability measure on $(\Omega ,\mathcal{F})$ is a [[Function|mapping]] $\mathbb{P}:\mathcal{F}\to \mathbb{R}$ satisfying Kolmogorov's axioms: > > (1): For all $B\in \mathcal{F},$ $\mathbb{P}(B)\in[0,1]$; > > (2): $\mathbb{P}(\Omega)=1$; > > (3) Countable additivity: For all sequences of pairwise disjoint events $(A_{i})_{i\geq 1},$ $\mathbb{P}\left(\bigcup_{n=1}^\infty A_n\right)=\sum_{n=1}^\infty\mathbb{P}(A_n)$where $\left(\bigcup_{n=1}^\infty A_n\right)$ is a [[Union of sets|union]] and $\sum_{n=1}^\infty\mathbb{P}(A_{n})$ is a [[Series of Real Sequence|series]]. # Properties See [[Uniform Probability Measure]]. By [[Probability of Set Difference of Events]], $\mathbb{P}(B-A)=\mathbb{P}(B)-\mathbb{P}(A\cap B)$ for all $A,B\in \mathcal{F}.$