Definition 1 Let . Define
Then is called d-dimensional Hausdorff measure in .
Remark 1 Hausdorff measure is a Borel outer measure, it is not Radon for because is not finite with respect to . And it is a measure if restricted on Lebesgue measurable sets (by Caratheodory condition: We say a set satisfies Caratheodory condition, if for any set , ). Moreover, note that gives the volume of unit ball in dimension if is an integer, we naturally have (not trivially) , where is the n-dimensional Lebesgue measure.
Here are some basic properties of Hausdorff measures:
Theorem 2 on , for all . And hence on .
Proof: Note that , we have for .
Denote by for . Then
So for all , hence on .
Theorem 3 on for .
Proof: Let be a unit cube in , we can divide it into cubes with side-length , diameter . So
Send we will have , hence on .
Theorem 4 Let , , then
- If , then .
- If , then .
Proof: Note that the second statement is the contraposition of the first one. To prove the first statement, fix a , there exists a cover of , with , such that
Then we have
Send we will have .
Having this theorem, we know that for any set there exists only one such that is neither nor . It is natural to define this to be its dimension.
Definition 5 We say the Hausdorff dimension of a set is
Note that the Hausdorff dimension need not be an integer. Even if is an integer, it doesn’t mean is a -dimensional surface in any sense.
Another important and related measure is called capacity, and it is defined as follows.
Definition 6 Let , we define the p-capacity of by
Obviously, by density we can consider instead of .
It’s not hard to see is an outer measure, by knowing that given a sequence of functions , denote by and , then -a.e. As we know, given an outer measure, one can construct a -algebra by Caratheodory condition such that it is a measure restricted to this -algebra. But in this case, we are not that interested in it because the -algebra only contains sets of capacity 0 or . And here are some properties of capacities that one can easily prove.
Theorem 7 (Properties of )
- , for .
From the scaling, we can see that the capacity does not measure the “volume” of the set. And according this scaling, it is natural to look for the relations between and .
Theorem 8 Assume , if , then .
Note that this is a refinement of the third statement of last theorem
Proof: First we claim that there exists a constant such that for any neighborhood containing , there exists an open set and such that
To proof the claim, let be a neighborhood of and set . Since , we can find a sequence of ball with such that , and
Then . Set and , then and
Now we use the claim inductively. We find a sequence of , from the claim, and relate them by . Set and . Then and on . Therefore
since are discrete and .
The next important relation is that, given , we will have some information on its Hausdorff dimension. Before proving it, we need a Lemma.
Lemma 9 Let , suppose and define
Proof: From Lebesgue differential theorem one can see . Then by the absolute continuity of Lebesgue integral, there exists a and such that whenever . Fix a and an , we define a set
Since , we can fix an open set with , and consider a family of balls
Then this family of balls cover , and by Vitali’s covering theorem, there exists a sequence of disjoint balls such that . Hence
Then we can take to yield the result.
Theorem 10 Let and . If , then for all .
Proof: Since , we can choose a sequence of functions such that . Define , by GNS inequality we know is well defined in , and for any . So , the average of on , goes to 0 as if . Now I claim that
Assume by contradiction, there exists an such that , by Poincare inequality, we know that
for small enough. Then
for some . This means converges as , which leads to a contradiction. Therefore, , and by the previous lemma we know since .
For case, we can say more about the relation between capacity and Hausdorff measure since we will have more geometric insights. For instance, we can take advantage of isoperimetric inequality and co-area formula to prove the following result.
Theorem 11 Let , then if and only if .
Proof: From Theorem 8 we know that implies . Assume , for any , we can find an such that and . By co-area formula, we know
This implies and for some . Fix this , by isoperimetric inequality, one can see . Obviously we know , then for each , we can find an such that
Since is a small portion in , by relative isoperimetric inequality we have
Since these balls cover , by Vitali’s covering theorem, there exists a sequence of disjoint balls such that . From the estimates we get above,
And we know that for each . Hence . Send we will get the result.
Remark 2 In the previous theorem, we actually prove the smallness of is equivalent to the smallest of , which we don’t have for .
The next important result is fine property of Sobolev functions. Comparing to Lebesgue integrable functions, Sobolev functions have a smaller singular set, and their representatives enjoy a better continuity result. To show it, we first need a Chebyshev type inequality.
Lemma 12 Let and , define the set
Proof: First note that, without loss of generality, we can assume by homogeneity. For each , we consider a ball such that . This collection of balls cover , and by Besicovitch covering theorem, there exist and countable collections of disjoint balls , such that
Denote by the element in for . For each , consider the function , which is in , on . By extension theorem we know this function can be extended to , say , such that
for all , where depends only on . Then we have in for all . Define , then since is open. Therefore
Now we are ready to prove the fine properties of Sobolev functions.
Theorem 13 Let . Then there exists a Borel set such that , and exists for any . In addition,
for all . Furthermore, given any , there exists an open set such that is continuous restricted on .
Proof: For each , choose an such that . Define the set
From Lemma 14, we know . Consider the set
then by Theorem 9 and Lemma 10, we have . For any , by Poincare’s inequality we have
Then for any , we have
Define , then for , we have . This implies converges uniformly to a continuous function in . So
as , which implies . Define , then
And we know that exists for all . In addition,
For the continuity statement, given any , we can find an such that . From theorem 8, we can find an open set such that . Then we are done since we know is continuous restricted on .