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
.
A point x is in the bdry of S if any neighborhood of x contains points in S and pts not in S …
Sorry, something was left out of a revision.. I’ll fix it to something that’s equivalent to what you just said, KEvin.
[…] . We must ask that be Jordan measurable, because this will happen if and only if the boundary has zero Jordan content, and thus zero outer Lebesgue measure. Since the collection of new discontinuities must be […]
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[…] notice how the last step of this proof echoes our earlier result that a set is Jordan measurable if and only if the Jordan content of its boundary is […]
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