product measure
A product measure is how you measure ‘area’ once you already know how to measure ‘length’ along each axis. If one space measures horizontal extent and another measures vertical extent, the product measure assigns to a rectangle the product of its width and its height, and then extends that rule consistently to far more complicated regions of the combined space. It is the abstract engine behind double integrals.
Given two sigma-finite measure spaces (X, A, mu) and (Y, B, nu), the product sigma-algebra A ⊗ B is generated by the measurable rectangles A × B with A in A and B in B. There is a unique measure mu × nu on this sigma-algebra satisfying (mu × nu)(A × B) = mu(A) times nu(B) for all such rectangles. Uniqueness needs sigma-finiteness; without it the construction can fail to be unique. The Lebesgue measure on R^2 arises exactly this way as the product of two copies of Lebesgue measure on R (after completing).
The point of building this measure carefully is to license integration over the product one variable at a time. The measure of a general set is computed by slicing: fix x, measure the slice in Y, then integrate those slice-measures over X — and the Fubini–Tonelli theorem certifies that doing it in the other order gives the same answer. A subtlety: the product of two complete measures (such as Lebesgue × Lebesgue) need not be complete, so one usually completes it afterward.
On X = Y = [0, 1] with Lebesgue measure, the product measure of the triangle T = {(x, y) : 0 <= y <= x <= 1} is computed by slicing: for fixed x the slice has length x, so (mu × nu)(T) = the integral over [0,1] of x dx = 1/2 — the area of the triangle.
Slicing turns a two-dimensional measure into nested one-dimensional integrals.