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Concept# Probability axioms

Summary

The Kolmogorov axioms are the foundations of probability theory introduced by Russian mathematician Andrey Kolmogorov in 1933. These axioms remain central and have direct contributions to mathematics, the physical sciences, and real-world probability cases. An alternative approach to formalising probability, favoured by some Bayesians, is given by Cox's theorem.
The assumptions as to setting up the axioms can be summarised as follows: Let be a measure space with being the probability of some event , and . Then is a probability space, with sample space , event space and probability measure .
The probability of an event is a non-negative real number:
where is the event space. It follows that is always finite, in contrast with more general measure theory. Theories which assign negative probability relax the first axiom.
Unitarity (physics)
This is the assumption of unit measure: that the probability that at least one of the elementary events in the entire sample space will occur is 1
This is the assumption of σ-additivity:
Any countable sequence of disjoint sets (synonymous with mutually exclusive events) satisfies
Some authors consider merely finitely additive probability spaces, in which case one just needs an algebra of sets, rather than a σ-algebra. Quasiprobability distributions in general relax the third axiom.
From the Kolmogorov axioms, one can deduce other useful rules for studying probabilities. The proofs of these rules are a very insightful procedure that illustrates the power of the third axiom, and its interaction with the remaining two axioms. Four of the immediate corollaries and their proofs are shown below:
If A is a subset of, or equal to B, then the probability of A is less than, or equal to the probability of B.
In order to verify the monotonicity property, we set and , where and for . From the properties of the empty set (), it is easy to see that the sets are pairwise disjoint and . Hence, we obtain from the third axiom that
Since, by the first axiom, the left-hand side of this equation is a series of non-negative numbers, and since it converges to which is finite, we obtain both and .

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