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:
“I had a feeling that out there, there were very poor people who didn’t have enough to eat. But they wore wonderfully colored rags and did musical numbers up and down the streets together.”
—Jill Robinson (b. 1936)
“Outward simplicity befits ordinary men, like a garment made to measure for them; but it serves as an adornment to those who have filled their lives with great deeds: they might be compared to some beauty carelessly dressed and thereby all the more attractive.”
—Jean De La Bruyère (1645–1696)