### 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.