measure-preserving transformation We say that the transformation
(or map) f preserves the measure
if the measure of a set A
is the same as the sum of the measures of all the pre-images of A
under f: that is,
. Since f may
be many-to-one, this is a different definition than would result if
we asked for
.
For the Gauss map
frac
, we can fairly easily show
that it preserves the Gauss measure
by considering the so-called basic sets
for
.
The pre-image of the set
is the set of intervals
for
. The measures of these intervals add up
to
Other sets in
may be formed by countable unions and
intersections of these sets, and hence the Gauss measure is preserved
by the Gauss map. The only difficult part of this computation is the
evaluation of the sum in the middle, and this is tedious but straightforward.
In passing I note that Maple can evaluate this sum, when properly coached.