the probability space (Omega, F, P)

The probability space is the single object that contains everything needed to do probability rigorously. It answers, in one package, the three questions any probabilistic model must settle: what can happen, which collections of happenings count as events, and how much probability each event gets. Every random variable, every expectation, every limit theorem in the subject is ultimately a statement about some probability space, even when it stays hidden in the background.

It is a triple (Omega, F, P). Omega is the sample space — the set of all possible outcomes of the experiment. F is a sigma-algebra of subsets of Omega — the family of events you are allowed to ask about, closed under complement and countable union. P is a probability measure on (Omega, F) — a countably additive set function with P(Omega) = 1 that assigns each event its probability. The first two pieces, (Omega, F), form the measurable space (the stage and the allowed questions); adding P (the assignment of likelihoods) completes it into a probability space. Reading the triple left to right: outcomes, then events, then probabilities.

The quiet genius of this setup, due to Kolmogorov in 1933, is that the same three-part skeleton handles a coin toss, a continuous measurement, and an infinite sequence of trials with no change in the rules — only the contents of Omega, F, and P differ. In practice the underlying space is often left unstated: we work directly with random variables and their distributions and trust that some probability space exists carrying them. Construction theorems (like Caratheodory and Kolmogorov extension) guarantee it does, so you rarely have to write Omega down.