The Measure Algebra of the Unit Interval
Let be the unit interval
, let
be the class of Borel sets on
, and let
be Lebesgue measure. If
is a sequence of partitions of the maximal element
of the measure algebra
into intervals, and if the limit of the sequence of norms
is zero, then
is dense.
If is a positive number, then we can find some
so that
. If
is a subinterval of
, then let
be the unique interval containing the left endpoint of
. If
also contains the right endpoint, then we can stop. Otherwise, let
be the next interval of
to the right of
, and keep going (at most a finite number of steps) until we get to an interval
containing the right endpoint of
. The union
of all the
can overshoot
by at most
on the left and the same amount on the right, and so
. Thus any interval can be approximated arbitrarily closely by some partition in the sequence. The general result follows, because we can always find a finite collection of intervals whose union is arbitrarily close to any given Borel set.
Now, say that every separable, non-atomic, normalized measure algebra is isomorphic to the measure ring
.
Since the metric space has a diameter of
— no two sets can differ by more than
, and
— we can find a dense sequence
in the space. For each
we can consider sets of the form
where either or
. The collection of all such sets for a given
is a partition
. It should be clear that the sequence
is decreasing, and the fact that
is dense implies that the sequence of partitions is also dense. We thus conclude that
.
The first partition has two elements
and
. We can define
and
, so that
and
. This gives us a partition
of
that reflects the structure of
. Similarly, we can carve up each of these intervals so that the resulting partition
reflects the structure of
. And we can continue, each time subdividing
into smaller intervals so that the next partition
reflects the structure of
. Since this correspondence preserves measures, it follows that
, and the above result then shows that the sequence
is dense.
Now, we can extend from partition elements occurring in
to finite unions of such elements by sending such a finite union to the finite union of corresponding elements of
. This gives an isometry from a dense subset of
to a dense subset of
. Thus we can uniquely extend it to an isometry
. Since
preserves unions and differences, and since these operations are uniformly continuous, it follows that
gives us an isomorphism of measure algebras.