measure zero

A set of measure zero is one so small, in the sense of length, that you can hide it under intervals of as little total length as you please. Pick any tiny budget epsilon greater than 0; you can still cover the entire set by intervals whose lengths add up to less than epsilon. Sets like this are negligible for the purposes of integration.

Precisely, a set N of real numbers has Lebesgue measure zero if for every epsilon greater than 0 there is a countable collection of intervals covering N with total length less than epsilon; equivalently, its Lebesgue outer measure is 0. Every countable set is null, but the converse fails: there are uncountable null sets. Measure zero is inherited by subsets and is preserved under countable unions, since a countable union of null sets is null.

Null sets are where measure theory hides its dirt: anything that happens only on a null set can usually be ignored. This is the meaning of “almost everywhere.” A surprising and important fact is that uncountability is no obstacle to being null — the Cantor set is uncountable, has the same cardinality as R, and yet has measure zero, dramatically separating “how many points” from “how much length.”