Some Sets of Measure Zero
Here’s a useful little tool:
Let be a countable collection of sets of measure zero in . That is, for all . We define to be the union
Then it turns out that as well.
To see this, pick some . For each set we can pick a Lebesgue covering of so that . We can throw all the intervals in each of the together into one big collection , which will be a Lebesgue covering of all of . Indeed, the union of a countable collection of countable sets is still countable. We calculate the volume of this covering:
where the final summation converges because it’s a geometric series with initial term and ratio . This implies that .
As an example, a set consisting of a single point has measure zero because we can put it into an arbitrarily small open box. The result above then says that any countable collection of points in also has measure zero. For instance, the collection of rational numbers in is countable (as Kate mentioned in passing recently), and so it has measure zero. The result is useful because otherwise it might be difficult to imagine how to come up with a Lebesgue covering of all the rationals with arbitrarily small volume.