Nowhere Dense Set - Nowhere Dense Sets With Positive Measure

Nowhere Dense Sets With Positive Measure

A nowhere dense set is not necessarily negligible in every sense. For example, if X is the unit interval, not only is it possible to have a dense set of Lebesgue measure zero (such as the set of rationals), but it is also possible to have a nowhere dense set with positive measure.

For one example (a variant of the Cantor set), remove from all dyadic fractions, i.e. fractions of the form a/2n in lowest terms for positive integers a and n, and the intervals around them: (a/2n − 1/22n+1, a/2n + 1/22n+1). Since for each n this removes intervals adding up to at most 1/2n+1, the nowhere dense set remaining after all such intervals have been removed has measure of at least 1/2 (in fact just over 0.535... because of overlaps) and so in a sense represents the majority of the ambient space . This set nowhere dense, as it is closed and has an empty interior: any interval (a, b) is not contained in the set since the dyadic fractions in (a, b) have been removed.

Generalizing this method, one can construct in the unit interval nowhere dense sets of any measure less than 1.