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Definition

Probability Space

Probability Theory · probability.tex
The triple $(\Omega, \mathcal{F}, P)$ is a probability space if
  1. $\Omega$ is the sample space, that is some possibly abstract set.
  2. $\mathcal{F}$ is a $\sigma$-algebra of sets - the measurable subsets of $\Omega$.
  3. $P$ is a probability measure.
that is, $P$ follows the Kolmogorov Axioms :
  1. For any $A \in \mathcal{F}$, there exists a number $P(A) \ge 0$, the probability of $A$.
  2. $P(\Omega) = 1$
  3. Let $\{A_n , n \ge 1\}$ be disjoint. Then $$ P\left(\bigcup_{n=1}^\infty A_n \right) = \sum_{n=1}^\infty P(A_n)
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