Axiom:Kolmogorov Axioms

Definition
Let $\EE$ be an experiment.

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

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:

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


 * $\forall A, B \in \Sigma: A \cap B = \O \implies \map \Pr {A \cup B} = \map \Pr A + \map \Pr B$

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

 * Elementary Properties of Probability Measure


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