Axiom:Kolmogorov Axioms

Definition

Let $\EE$ be an experiment.

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability measure on $\EE$.

Then $\EE$ can be defined as being a measure space $\struct {\Omega, \Sigma, \Pr}$, such that $\map \Pr \Omega = 1$.

Thus $\Pr$ satisfies the Kolmogorov axioms:

$(1)$	$:$	$\ds \forall A \in \Sigma:$	$\ds 0 $	$\ds \le $	$\ds \map \Pr A \le 1 $	The probability of an event occurring is a real number between $0$ and $1$
$(2)$	$:$		$\ds \map \Pr \Omega $	$\ds = $	$\ds 1 $	The probability of some elementary event occurring in the sample space is $1$
$(3)$	$:$		$\ds \map \Pr {\bigcup_{i \mathop \ge 1} A_i} $	$\ds = $	$\ds \sum_{i \mathop \ge 1} \map \Pr {A_i} $	where $\set {A_1, A_2, \ldots}$ is a countable (possibly countably infinite) set of pairwise disjoint events
						That is, the probability of any one of countably many pairwise disjoint events occurring
						is the sum of the probabilities of the occurrence of each of the individual events

Some sources include:

$\map \Pr \O = 0$

but this is strictly speaking not axiomatic as it can be deduced from the other axioms.

Definition:Measure Space: the Kolmogorov axioms follow directly from the fact that $\struct {\Omega, \Sigma, \Pr}$ is an example of such.

\((1)\)	$:$	\(\ds \forall A \in \Sigma:\)	\(\ds 0 \)	\(\ds \le \)	\(\ds \map \Pr A \le 1 \)	The probability of an event occurring is a real number between $0$ and $1$
\((2)\)	$:$		\(\ds \map \Pr \Omega \)	\(\ds = \)	\(\ds 1 \)	The probability of some elementary event occurring in the sample space is $1$
\((3)\)	$:$		\(\ds \map \Pr {\bigcup_{i \mathop \ge 1} A_i} \)	\(\ds = \)	\(\ds \sum_{i \mathop \ge 1} \map \Pr {A_i} \)	where $\set {A_1, A_2, \ldots}$ is a countable (possibly countably infinite) set of pairwise disjoint events
						That is, the probability of any one of countably many pairwise disjoint events occurring
						is the sum of the probabilities of the occurrence of each of the individual events