Probability axioms

Probability axioms define a probability measure: event probabilities are nonnegative, the entire sample space has probability one, and disjoint events have additive probabilities.

Sample space: The set of all possible outcomes in a probabilistic model. The unit-probability axiom assigns probability one to this set.
Countable additivity: The rule that probabilities of countably many disjoint events sum to the probability of their union. It extends finite additivity to infinite collections of disjoint events.
Andrey Kolmogorov: A Soviet mathematician whose 1933 work established probability theory on measure-theoretic foundations. His axiomatization became the standard formal basis for modern probability.
Frequentist probability: An interpretation of probability as the long-run relative frequency of outcomes under repeated trials. It interprets probabilities constrained by the axioms through repeated experimental outcomes.
Event (probability theory): A measurable subset of a sample space whose probability can be evaluated. The axioms assign probabilities to events, not merely individual outcomes.
Complement rule: The identity that an event's probability and its complement's probability sum to one. It follows from unit probability and additivity for an event and its complement.
Foundations of the Theory of Probability: Andrey Kolmogorov's 1933 book presenting a measure-theoretic axiomatization of probability. This work set out the framework in which the familiar axioms became canonical.
Bayesian probability: An interpretation of probability as a coherent degree of belief, updated with evidence. It uses the same probability rules to represent uncertainty about hypotheses.
Probability measure: A function assigning numbers to measurable sets while satisfying probability axioms. The axioms are precisely the conditions that make this function a probability measure.
Continuity of probability measures: Limit properties describing probabilities of increasing or decreasing sequences of events. Countable additivity entails these useful continuity laws.

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