a probability measure
A probability measure is just a measure with the total size set to exactly 1 — it is the rigorous form of the rules of probability you already know. Instead of measuring length or mass, it measures how much belief, weight, or long-run frequency sits on each event, normalized so that something certain has weight 1. Everything you learned as Kolmogorov's axioms — probabilities are non-negative, the whole sample space has probability 1, and disjoint events add — is exactly the statement that P is a measure with P(Omega) = 1.
Formally, a probability measure P on a measurable space (Omega, F) assigns to each event A a number P(A) in [0, 1] with P(Omega) = 1 and the countable-additivity property: for disjoint events A1, A2, A3, ..., P of their union equals P(A1) + P(A2) + P(A3) + .... From these two requirements every familiar rule drops out: P(empty set) = 0, the complement rule P(not A) = 1 - P(A), monotonicity (if A is inside B then P(A) <= P(B)), inclusion-exclusion, and continuity from below and above. The key upgrade over elementary probability is that additivity holds for countably many events, not just finitely many — and that single change is what lets you take limits of events, define tail events, and prove the laws of large numbers.
Why insist on countable additivity? Because almost everything interesting in probability is a limit: the event that a sequence converges, the long-run fraction of heads, the probability a random walk ever returns to the origin. Mere finite additivity cannot reach these. The price is that P is defined only on the sigma-algebra F, not on every subset — but that price is unavoidable, since non-measurable sets show some subsets simply cannot be given a consistent probability.
On Omega = {1,...,6} with all subsets as events, the fair-die probability measure sets P({k}) = 1/6 for each face, and additivity gives P({even}) = P({2}) + P({4}) + P({6}) = 1/2. On (R, Borel sets), Normal(0,1) is a probability measure where P([a,b]) is the area under the bell curve from a to b.
A measure normalized so the whole space weighs 1 — Kolmogorov's axioms, stated as a single measure-theoretic object.
A probability measure is defined only on the events in F, never on arbitrary subsets; asking for P of a non-measurable set is meaningless, not merely unknown.