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.
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