measure
A measure is a precise way of assigning a “size” — a length, an area, a volume, a probability — to sets, in a way that behaves the way sizes ought to. If you cut a region into non-overlapping pieces, the sizes of the pieces should add up to the size of the whole. A measure is the mathematical machine that does exactly this, and crucially it does so even for infinitely many pieces.
Formally, fix a set X and a sigma-algebra A of subsets of X (the “measurable” sets). A measure is a function mu from A to the extended non-negative reals [0, +infinity] such that mu(empty set) = 0 and mu is countably additive: for any sequence of pairwise disjoint sets A_1, A_2, ... in A, mu(union of the A_n) = sum of mu(A_n). The triple (X, A, mu) is called a measure space.
Countable additivity is the heart of the matter and is exactly what makes measure theory more powerful than naive area. It is strictly stronger than finite additivity: finitely additive set functions that are not countably additive do exist, but they are too weak to support a good theory of integration. Note also that a measure is allowed to take the value +infinity (the whole real line under Lebesgue measure), and that two different measurable sets can have the same measure.
On the real line with Lebesgue measure m, the interval [0, 1] has m([0,1]) = 1, a single point {3} has m({3}) = 0, and the whole line has m(R) = +infinity. Splitting [0, 2] into [0, 1) and [1, 2] gives 1 + 1 = 2, as additivity demands.
A measure adds up over disjoint pieces — even infinitely many.
From countable additivity one derives basic properties for free: monotonicity, finite additivity, and continuity from below and (with a finiteness caveat) from above. These are the daily tools of the subject.