Lebesgue measure

Lebesgue measure is the rigorous, all-purpose generalization of length on the line, area in the plane, and volume in space. It agrees with the elementary notions where those make sense — the measure of [a, b] is b minus a — but it extends them to a vast collection of complicated sets that have no obvious length, all while never moving when you slide the set around.

On the real line, Lebesgue measure m is the unique measure, defined on the Lebesgue sigma-algebra, that assigns to every interval its ordinary length and is translation invariant: m(E + x) = m(E) for every measurable E and every real x. It is constructed by restricting Lebesgue outer measure to the sets satisfying Carathéodory’s criterion. In R^n the same construction yields n-dimensional volume, again translation invariant and agreeing with elementary volume on boxes.

Lebesgue measure is the canvas on which the Lebesgue integral is painted, and it is what makes phrases like “almost everywhere” precise. Two honest caveats: it is not defined on every subset of R (non-measurable sets exist), and no translation-invariant measure on all subsets of R can assign length b minus a to intervals — the Vitali construction shows this is impossible, which is precisely why a sigma-algebra is needed.