Probability Measure

Topics: Probability

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(definition)

A probability measure (or probability function or simply probability) is a function $\mathbb{P}:\mathcal{F}\to [0,1]$ ($\mathcal{F}$ a σ-algebra) such that:

1. $\mathbb{P}(A)\ge 0$ (non-negative)
2. $\mathbb{P}(\Omega )=1$
3. If $(A_n{)}_{n\ge 1}\in \mathcal{F}$ such that $A_i\cap A_j=\varnothing$ ($i\ne j$), then:

$$\mathbb{P}\left(\bigcup _{n=1}^{\infty }{A}_n\right)=\sum _{n=1}^{\infty }\mathbb{P}(A_n)$$

Probability measures have specific properties.