A set X is said to be nowhere dense if the interior of the set closure of X is the empty set. For example, the Cantor set is nowhere dense. There exist nowhere dense sets of positive measure. For example, enumerating the rationals in [0, 1] as {q_n} and choosing an open interval I_n of length 1/3^n containing q_n for each n, then the union of these intervals has measure at most 1/2. Hence, the set of points in [0, 1] but not in any of {I_n} has measure at least 1/2, despite being nowhere dense.

Baire category theorem | dense | measure zero | positive measure