Axiom:Kolmogorov Axioms

From ProofWiki
Jump to navigation Jump to search

Definition

Let $\mathcal E$ be an experiment.

Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability measure on $\mathcal E$.


Then $\mathcal E$ can be defined as being a measure space $\left({\Omega, \Sigma, \Pr}\right)$, such that $\Pr \left({\Omega}\right) = 1$.


Thus $\Pr$ satisfies the Kolmogorov axioms:


Axioms

First Axiom

$\forall A \in \Sigma: 0 \le \Pr \left({A}\right) \le 1$

The probability of an event occurring is a real number between $0$ and $1$.


Second Axiom

$\Pr \left({\Omega}\right) = 1$

The probability of some elementary event occurring in the sample space is $1$.


Third Axiom

Let $A_1, A_2, \ldots$ be a countable (possibly countably infinite) sequence of pairwise disjoint events.

Then:

$\displaystyle \Pr \left({\bigcup_{i \mathop \ge 1} A_i}\right) = \sum_{i \mathop \ge 1} \Pr \left({A_i}\right)$


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.


As an elementary and easily-digested consequence of this, we have:

$\forall A, B \in \Sigma: A \cap B = \varnothing \implies \Pr \left({A \cup B}\right) = \Pr \left({A}\right) + \Pr \left({B}\right)$


Also defined as

Some sources include:

$\Pr \left({\varnothing}\right) = 0$

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 $\left({\Omega, \Sigma, \Pr}\right)$ is an example of such.


Source of Name

This entry was named for Andrey Nikolaevich Kolmogorov.