Axiom:Kolmogorov Axioms

From ProofWiki
Jump to navigation Jump to search


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

Also defined as

Some sources include:

$\map \Pr \O = 0$

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

Also see

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

Source of Name

This entry was named for Andrey Nikolaevich Kolmogorov.