Subsets of the real line of questionable length

Kate Okikiolu
Johns Hopkins University

Using a ruler, we can measure the length of an interval in the real line. If we have a set which is the union of a finite number of disjoint intervals in the real line then the measure of this set is the sum of the lengths of these intervals. But why stop at sets of this kind?
If we have a set which is a countable union of disjoint intervals in the real line, then we can define its measure to be the sum of the lengths of these intervals. This measure is well defined.
But why stop here? What about a general set?
This is an old and well studied problem from which many interesting questions still arise.


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