Probability space

A probability space is a mathematical triplet (Ω, F, P) consisting of a sample space Ω of possible outcomes, a σ-algebra F of events, and a probability measure P that assigns probabilities to events. It is the stage on which random variables perform.

The probability space formalizes what it means for something to be "possible" and "probable." The σ-algebra F determines which questions can be asked: not every subset of outcomes need be measurable. This restriction is not a limitation but a necessity — without it, paradoxical sets can be constructed that defy consistent probability assignment. The Banach-Tarski paradox is the ghost that haunts unrestricted measure assignment. The probability space is where measure theory meets interpretation, and the choice of σ-algebra is as consequential as the choice of measure itself.