Kolmogorov axioms

The Kolmogorov axioms are an axiomatization of probability theory.

Axioms

Let $P:\Sigma \to [0,1]$ be function such that $\Sigma$ is a sigma-algebra on the set $\Omega$. Then $P$ is a probablity measure if the following holds:

1. For all $E \in \Sigma$ $P(E)\geq 0$
2. $P(\Omega)=1$
3. For all $A, B\in \Sigma$, if $A \cap B = \emptyset$ then $P(A \cup B)=P(A)+P(B)$

The ordered pair $(\Omega,\Sigma,P$) is called a probablity space. Here, $\Omega$ is called the Sample space, $\Sigma$ is a collection of events, and $P$ is said to be the probablity measure.

It follows by induction on statement 3 that: $P(\cup^{n}_{i=1} A_{i})= \sum^n_{i=1} P(A_i)$

Important Consequences

• Complement Rule
• Addition Theorem of Probablity