Probability measure

A probability measure is a function $\mathbb{P}:\mathcal{F}\to[0,1]$ defined on a sigma-algebra $\mathcal{F}$ of subsets of a set $\Omega$ such that $\mathbb{P}(\varnothing)=0$, $\mathbb{P}\!\left(\bigcup_{n=1}^\infty A_n\right)=\sum_{n=1}^\infty \mathbb{P}(A_n)$ for every pairwise disjoint sequence $(A_n)_{n\ge1}\subseteq\mathcal{F}$, and $\mathbb{P}(\Omega)=1$.

A probability measure is a special case of a measure and is the key ingredient in a probability space ; it assigns probabilities to events (measurable sets).

Examples:

• If $\Omega=\{1,2,\dots,n\}$ and $\mathcal{F}=2^\Omega$, the uniform probability measure is $\mathbb{P}(A)=|A|/n$.
• If $\Omega=[0,1]$, $\mathcal{F}$ is the Borel $\sigma$-algebra, and $\lambda$ is Lebesgue measure , then $\mathbb{P}(A)=\lambda(A)$ defines the uniform probability measure on $[0,1]$.