Lebesgue measure and nonmeasurable sets

Lebesgue measure on the real line is a complete, translation-invariant measure agreeing with interval length. Every open set is a countable union of disjoint open intervals, and all Borel sets are measurable. Completing Borel measure adds every subset of every Borel null set. More precisely, each Lebesgue measurable set differs from a Borel set by a subset of a Borel null set. This description allows some functions that are not Borel measurable to be Lebesgue measurable after null-set modification.

Several notions of size remain independent. The rational numbers are dense but null because they are countable. The complement of the rationals in any interval has full measure in that interval but contains no open subinterval consisting entirely of irrational points. Countable additivity does not apply to the uncountable union of singleton points making up an interval. Each singleton having measure zero therefore does not force every set to have measure zero. The quantifier countable is doing essential work.

Construct the middle-thirds Cantor set by removing the middle third of each surviving interval repeatedly. After n stages, the remaining set Cₙ consists of 2ⁿ closed intervals of length 3⁻ⁿ. Its total length is therefore (2/3)ⁿ. The final intersection is closed and bounded, hence compact, and lies inside every Cₙ. Monotonicity gives its measure at most every one of these lengths. Taking n arbitrarily large proves that the Cantor set is null without requiring a list of its points.

To see uncountability, associate each infinite binary sequence with the ternary number whose digits are twice those binary digits. The resulting numbers lie in the Cantor set, and distinct binary sequences produce distinct numbers under this restricted digit construction. Thus uncountably many choices fit inside a set of measure zero. The endpoints' alternative ternary descriptions do not destroy this injection when one uses the all-zero-or-two representation consistently. A finite stage has positive length; the limit set has zero length.

On [0,1], declare two points equivalent when their difference is rational. Using the axiom of choice, select one representative from each equivalence class and call the selection V. Its rational translates V+q for rational q in [−1,1] are pairwise disjoint: an overlap would identify two selected representatives of the same class, forcing them equal and then forcing the translation amounts equal. The union of these translates covers [0,1] and is contained in [−1,2]. Both containment statements matter to the contradiction.

If V were measurable with measure zero, countable additivity would give measure zero to that union, contradicting its covering [0,1]. If V had positive measure, infinitely many disjoint translates would give the bounded union infinite measure, contradicting containment in [−1,2]. Infinite measure for V is already impossible since V lies in [0,1]. The conclusion is failure of measurability, not failure of outer measure to exist. The construction uses choice and the rational equivalence relation; an arbitrary complicated-looking subset is not automatically nonmeasurable.