Measure Zero & Lebesgue Criterion

Negligibly small made precise, and exactly which functions integrate.

The idea

A set has measure zero when it can be covered by boxes of arbitrarily small total volume; these are the sets small enough for integration theory to ignore.

Definition (Measure zero).

A set $S \subseteq \mathbb{R}^{n}$ has measure zero if for every $\varepsilon > 0$ there are countably many boxes $B_{1}, B_{2}, \dots$ with $S \subseteq \bigcup_{k} B_{k} \qquad \text{and} \qquad \sum_{k} \text{vol}(B_{k}) < \varepsilon.$

The definition permits countably many boxes rather than finitely many, and this is what lets an infinite set qualify: cover the $k$-th point of a countable set by a box of volume $\varepsilon 2^{-k}$, and the total volume is at most $\varepsilon$. So every countable set — the rationals included — has measure zero. Only the total volume is constrained, so a set spread across the whole line can still be negligible. In the plane the contrast is visible: a curve can be caught by a chain of thin boxes of total area less than $\varepsilon$, while a disk cannot — its own area is a floor that no covering's total can undercut.

The Lebesgue criterion uses this notion to say exactly how much discontinuity Riemann integration tolerates.

Theorem (Lebesgue criterion).

A bounded function on a box in $\mathbb{R}^{n}$ is Riemann integrable if and only if its set of discontinuities has measure zero.

The dividing line is not finitely many discontinuities, and not isolated ones, but measure zero precisely.

Ways to work on it

• Walkthrough. The definition of measure zero and the Lebesgue criterion for integrability.
• Practice. Decide whether a given subset of the line or plane has measure zero.
• Hardest. Apply the Lebesgue criterion to borderline integrable and non-integrable functions.

Not sure where to start? Take the ten-question placement test.