Borel Sets and Lebesgue Measure
Let’s consider some of the easy properties of the Borel sets and Lebesgue measure we introduced yesterday.
First off, every countable set of real numbers is a Borel set of measure zero. In particular, every single point is a Borel set. Indeed, can be written as the countable intersection
so it’s a Borel set. Further, monotonicity tells us that
and so the singleton has measure zero. But is countably additive, so given any countable collection the measure is the sum of the measures of the individual points, each of which is zero.
Next, as I said when I introduced semiclosed intervals, we could have started with open intervals, but the details would have been messier. Now we can see that the -ring generated by the collection of semiclosed intervals is the same as that generated by the collection of all open sets.
We can see, in particular, that each open interval is a Borel set. Indeed, the point is a Borel set, as is the semiclosed interval , and we have the relation . Every other open set in is a countable union of open intervals, and so they’re all Borel sets as well. Conversely, we could write
and find the singleton in the -ring generated by . Then we can write and find every semiclosed interval in this -ring as well. And thus
We can also tie our current measure back to the concept of outer Lebesgue measure we introduced before. Back then, we defined the “volume” of a collection of open intervals to be the sum of the “volumes” of the intervals themselves. We defined the outer measure of a set to be the infimum of the volumes of finite open covers. And, indeed, this is exactly the outer measure corresponding to Lebesgue measure .
Remember that the outer measure is defined for a set by
Since , we have the inequality
On the other hand, if is any positive number, then by the definition of we can find a sequence of semiclosed intervals so that
We can thus widen each of these semiclosed intervals just a bit to find
Since was arbitrary, we find that . And, thus, that
In effect, we’ve replaced the messily-defined “volume” of an open cover by the more precise Lebesgue measure , but the result is the same. The “outer Lebesgue measure” from our investigations of multiple integrals is the same as the outer measure induced by our new Lebesgue measure.