How did Andrey Kolmogorov help establish the modern axioms of probability?

Andrey Kolmogorov established the modern axiomatic foundation for probability theory by applying the principles of measure theory to the field. In his influential 1933 monograph, Grundbegriffe der Wahrscheinlichkeitsrechnung (Foundations of the Theory of Probability), he formalized probability using three core components:12

• Sample Space ($S$): The set of all possible outcomes of an experiment.2
• $\sigma$-field ($M$): A distinguished class of subsets of $S$ (called events) that satisfies specific closure properties, such as containing the sample space and being closed under complementation and countable unions.2
• Probability Measure ($P$): A function that assigns a non-negative real number to each event in $M$, satisfying two primary conditions: $P(S)=1$ and the axiom of countable additivity, which states that the probability of a countable union of disjoint events is the sum of their individual probabilities.2

By framing probability in these terms, Kolmogorov provided a rigorous mathematical structure that moved beyond finite sample spaces, allowing the theory to be applied consistently to complex problems in physics, biology, and stochastic processes.12