measure-theory Mathematics

A set is null if it can be covered by countably many intervals of arbitrarily small total length. Single points and all countable sets are null, and even the uncountable Cantor ternary set has measure zero. A statement holds almost everywhere when it fails only on some null set — monotone functions, for example, are differentiable almost everywhere. The Lebesgue integral is insensitive to values on null sets: changing a function there alters no integral, and functions differing only on a null set count as identical. This convention discards pathological exceptions and makes the integral limit theorems under pointwise convergence possible.

Riemann integration requires the oscillation to be small on every subinterval, yet every nonempty subinterval of [0,1] contains both rational and irrational points, so upper and lower sums never agree and the Riemann integral does not exist. Lebesgue integration instead stratifies by values: the function equals 1 on the rationals and 0 elsewhere, and since the rationals are countable they have Lebesgue measure zero, giving 1×0 + 0×1 = 0. This displays the theory's power over extremely discontinuous functions.

Lebesgue's criterion states that a bounded function on [a,b] is Riemann integrable if and only if its discontinuities form a set of Lebesgue measure zero. The proof covers points where oscillation exceeds any threshold by intervals of arbitrarily small total length, controlling the gap between Darboux upper and lower sums. Integrability thus becomes a measure question: monotone functions have at most countably many jumps and always integrate, while everywhere-discontinuous functions cannot.