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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1971 Springer-Verlag New York
About this chapter
Cite this chapter
Oxtoby, J.C. (1971). Measure and Category on the Line. In: Measure and Category. Graduate Texts in Mathematics, vol 2. Springer, New York, NY. https://doi.org/10.1007/978-1-4615-9964-7_1
Download citation
DOI: https://doi.org/10.1007/978-1-4615-9964-7_1
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-05349-3
Online ISBN: 978-1-4615-9964-7
eBook Packages: Springer Book Archive