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:
“Old age equalizes—we are aware that what is happening to us has happened to untold numbers from the beginning of time. When we are young we act as if we were the first young people in the world.”
—Eric Hoffer (1902–1983)
“What is life but the angle of vision? A man is measured by the angle at which he looks at objects. What is life but what a man is thinking all day? This is his fate and his employer. Knowing is the measure of the man. By how much we know, so much we are.”
—Ralph Waldo Emerson (1803–1882)