Measure and Category pp 1-5 | Cite as

# 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 (≦2*k* - 1) of rational numbers *p*/*q*in 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.

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