Probability Measure

A probability measure (denoted $P$) is a function that assigns a number in $[0,1]$ to each event in a Sigma-Algebra $\mathcal{F}$, satisfying:

1. $P(S)=1$
2. For any countable collection of mutually exclusive events $A_1,B_2,\dots$ in $\mathcal{F}$,

$$P\left(\bigcup _{i=1}^{\infty }{A}_i\right)=\sum _{i=1}^{\infty }P(A_i)$$

Interpretation:

• $P(A)$ quantifies the likelihood that event $A$ occurs.
• $P$ only applies to sets in the Sigma-Algebra $\mathcal{F}$.

The probability measure if the third component of Probability Space.