Theorem L

The map is a continuous, one-one, onto, and measure-preserving map on the 2-adic integers .

Proof L

This is essentially a consequence of Theorem B. Use the fact that . For any in , if , then , so

so that and is continuous. If , then for some n, so that

and is one-to-one. To see that is onto, given one can find so that

since is a permutation. Then . Now forms a Cauchy sequence in the 2-adic metric and is compact, hence the limiting value of satisfies . Now is defined, and implies that is continuous.