Liouville Number - Liouville Numbers and Measure

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.