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Traditional probability is based on Kolmogorov's three axioms of probability. Because they are axioms they don't require a proof and although they are intuitive I am wondering if there is a more rigorous way to be convinced of their validity other than intuition.

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    $\begingroup$ You could treat them as a definition of a probability measure, in contrast to other measures which might not necessarily sum to $1$ or always be non-negative, or without countable disjoint additivity. $\endgroup$ – Henry Jan 13 '17 at 19:56
  • $\begingroup$ The book by Dubins & Savage: How to gamble if you must. explores a probability system in which $P(A \cup B) = P(A) + P(B)$ for disjoint $A,B,$ but in which countable additivity is not included among the axioms. $\endgroup$ – BruceET Jan 13 '17 at 23:46
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Think of the properties of relative frequency. The first two axioms then are trivial if we consider the probabilities to be the counterparts of relative frequencies.

The third axiom is also trivial in the mirror of relative frequencies if we talk only about a finite number of pairwise excluding events.

The theory requires the generalization for an infinite number of such events. This cannot be made more intuitive.

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