Liouville Numbers and Measure
From the point of view of measure theory, the set of all Liouville numbers is small. More precisely, its Lebesgue measure is zero. The proof given follows some ideas by John C. Oxtoby.
For positive integers and set:
- – we have
Observe that for each positive integer and, we also have
Since and we have
Now and it follows that for each positive integer, has Lebesgue measure zero. Consequently, so has .
In contrast, the Lebesgue measure of the set of all real transcendental numbers is infinite (since is the complement of a null set).
In fact, the Hausdorff dimension of is zero, which implies that the Hausdorff measure of is zero for all dimension . Hausdorff dimension of under other dimension functions has also been investigated.
Read more about this topic: Liouville Number
Famous quotes containing the words numbers and/or measure:
“All ye poets of the age,
All ye witlings of the stage,
Learn your jingles to reform,
Crop your numbers to conform.
Let your little verses flow
Gently, sweetly, row by row;
Let the verse the subject fit,
Little subject, little wit.
Namby-Pamby is your guide,
Albion’s joy, Hibernia’s pride.”
—Henry Carey (1693?–1743)
“Unless a group of workers know their work is under surveillance, that they are being rated as fairly as human beings, with the fallibility that goes with human judgment, can rate them, and that at least an attempt is made to measure their worth to an organization in relative terms, they are likely to sink back on length of service as the sole reason for retention and promotion.”
—Mary Barnett Gilson (1877–?)