a measure
A measure is the precise, grown-up version of the everyday idea of size — length, area, volume, mass, or count. It is a rule that takes a set and returns a non-negative number (possibly infinity) saying how big it is, in a way that respects the obvious common-sense law: if you split a region into separate pieces that do not overlap, the size of the whole equals the sum of the sizes of the pieces. Length on the line, area in the plane, and ordinary mass are all measures; probability is the special case where the total size is exactly 1.
Formally, a measure mu on a measurable space (Omega, F) assigns to each measurable set A a value mu(A) in [0, infinity] such that mu(empty set) = 0 and mu is countably additive: for any countable collection of disjoint measurable sets A1, A2, A3, ..., the measure of their union equals the sum mu(A1) + mu(A2) + mu(A3) + .... That single countable-additivity rule is the engine of the whole theory — it is what lets you compute the size of a complicated set by chopping it into infinitely many simple pieces and adding. From it follow monotonicity (bigger sets have bigger measure) and continuity (the measure of an increasing chain of sets approaches the measure of their union). The most important example is Lebesgue measure, the unique measure on the Borel line that gives every interval [a, b] its ordinary length b - a.
Measures unify a surprising range of ideas under one definition. Counting measure (which simply counts the elements of a set) turns sums into integrals; Lebesgue measure makes the Lebesgue integral possible; a probability measure makes expectations rigorous. Because the framework is identical, theorems proved once for general measures apply at no extra cost to length, to counting, and to probability alike — which is precisely why measure theory became the common language of analysis and probability.
Lebesgue measure on the real line gives the interval [2, 5] measure 3 and the single point {7} measure 0. Because it is countably additive, the set of all rationals — a countable union of single points — has measure 0 + 0 + 0 + ... = 0, even though it is dense.
Countable additivity lets a measure size up an infinitely intricate set by adding up the sizes of its pieces.
Countable additivity is strictly stronger than finite additivity and is what makes limits behave; a merely finitely-additive set function is not a measure and breaks the convergence theorems probability relies on.