Jordan Content and Boundaries
As a first exercise working with Jordan content, let’s consider how it behaves at the boundary of a region.
I’ve used this term a few times when it’s been pretty clear from context, but let me be clear. We know about the interior and closure of a set, and particularly of a subset of . The boundary of such a set will consist of all the points in the closure of the set, but not in its interior. That is, we have
That is, a point is in if , but any neighborhood of contains points not in .
So let’s put inside a box and partition the box with a partition . When we calculate , we include all the subintervals that we do when we calculate , along with some other subintervals which contain both points within and points not in . I say that each of these are exactly the subintervals which contain points in . Indeed, if a subinterval contains a point of it cannot be included when calculating , but must be included when calculating . Inversely, if a subinterval contains no point of then it is either contained completely within — and is included in both calculations — or it is contained completely within the complement of — and is contained in neither computation. Thus we have the relation
which we rewrite
We can then pass to infima to find
Check this carefully to see how the equality is weakened to an inequality.
On the other hand, given any we can pick partitions so that and so that . We let be a common refinement of and , which will then satisfy both of these inequalities. We find
since is arbitrary, we find .
Together with the previous result, we conclude that . In particular, we find that is Jordan measurable if and only if .