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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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