Measure and Category on the Line

Abstract

The notions of measure and category are based on that of countability. Cantor’s theorem, which says that no interval of real numbers is countable, provides a natural starting point for the study of both measure and category. Let us recall that a set is called denumerable if its elements can be put in one-to-one correspondence with the natural numbers 1, 2, …. A countable set is one that is either finite or denumerable. The set of rational numbers is denumerable, because for each positive integer k there are only a finite number (≦2k - 1) of rational numbers p/qin reduced form (q > 0, p and q relatively prime) for which |p| + q = k. By numbering those for which k = 1, then those for which k = 2, and so on, we obtain a sequence in which each rational number appears once and only once. Cantor’s theorem reads as follows.