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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) \ge 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

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