Probability Theory: Kolmogorov’s Framework

The mathematical theory of probability arose from correspondence between Blaise Pascal (1623-1662) and Pierre Fermat (1601-1665) to solve some problems in the games of chance. The Kolmogorov’s Framework (KF) is based on the following axioms:

Definition 1 (Borel Sigma-Field): Let $\Omega$ and let $\mathcal{F}$ be a collection of subsets of $\Omega$ . Then $\mathcal{F}$ is called a Borel sigma-field if the following conditions are met:

1. The empty set $\varphi \in \mathcal{F}$ . 2. If $A$ in $\mathcal{F}$ , the complement $A^c$ in \mathcal{F}. 3. If $A\in\mathcal{F}$ for all $i =1,2,3,\ldots$ , then $\cup_{i=1}^\infty A_i\in\mathcal{F}$ .

Definition 2 (Probability Measure): A set function $P: \mathcal{F}\to [0,1]$ is said to be a probability measure if the following conditions hold:

1. $P(\Omega)=1$ , 2) $P(A)\geq 0$ , $A\in\mathcal{F}$ , and 3) $P(\cup_{i=1}^\infty)=\sum_{i=1}^\infty P(A)$ $A_i\in\mathcal{F}$ , $A_i\cap A_j=\phi$ , $i\not= j$ .

Definition 3 (Conditional Probability): Let $(\Omega, \mathcal{F}, P)$ be a probability space. Then the conditional probability of the event $A$ given that $B$ has occurred is defined as $P(A|B)=\frac{P(A\cup B)}{P(B)}, \ \ P(B)>0.$

Definition 4 (Random Variable): Let $w\in \Omega$ . Let $B\subset R^\ast =[-\infty,\infty]$ . Then the real-valued function $X: \Omega\to R^\ast$ is called a random variable if the image of $B$ under the inverse mapping $X^{-1}(B)=\{w: X(w)\in B\}\in\mathcal{F}$ , where $\mathcal{F}$ is a sigma-algebra on $\Omega$ .