Some remarks on the Hausdorff measure of the Cantor set

In this paper, the author further reveals some intrinsic properties of the Cantor set. By erties, the author gives a new method for calculating the exact value of the Hausdorff measure of the C and shows the facts that each covering which consists of basic intervals, in which any basic interval can pletely contain the other, is the best covering of the Cantor set, and the Hausdorff measure of the Cant be determined by coverings which only consist of basic intervals.


INTRODUCTION
It is well known that it is one of the most important subjects to calculate or estimate Hausdorff measures of fractal sets in fractal geometry.Generally speaking, it is very difficult to calculate Hausdorff measures of fractal sets, as Falconer said in [1]: "It is already becoming apparent that calculation of Hausdorff measures and dimensions can be a little involved, even for simple sets.Usually it is the lower estimate that is awkward to obtain."So far, there are few concrete results about computation of Hausdorff measures even for some simple fractals.The Hausdorff measure of the Cantor set which equals 1 is shown in [2].In [3], Ayer and Strichartz gave an algorithm for computing the Hausdorff measures of a class of Cantor sets.With the help of the principle of mass distribution, the Hausdorff measure of a Sierpinski carpet which equals 2 is obtained in [4].The references [5] and [6]  respectively gave the upper and lower bound for the Hausdorff measure of the Sierpinski gasket.
As what the authors pointed out in [7], the reason why it is so difficult to calculate the Hausdorff measure is neither computational trickiness nor computational capacity, but a lack of full understanding of the essence of the Hausdorff measure and a deep insight for the intrinsic properties of fractal sets.In this paper, we will further reveal some intrinsic properties of the Cantor set about Hausdorff measure.We show that each covering that consists of basic intervals in which any basic interval cannot completely contain the other is a best covering, and the Hausdorff measure of the Cantor set can be determined by coverings which only consist of basic intervals.It follows that a new method calculating the Hausdorff measure of the Cantor set is given.

PRELIMINARIES
Let U be a non-empty subset of n R , which is denoted by U , the diameter of U .If for each i , we say that ^ì U is a G -covering of E .Let E be a subset of n R and s be a non-negative number, for 0 !G , it is defined as follows: Where, the infimum is over all (countable) G -coverings ^ì U of E .The Hausdorff s-dimensional measure of E is defined as follows: is defined in a similar manner to the s-dimensional measure of E but using a restricted class N of covering sets in definition rather than the class of all sets.We say that the net 1 . It is well known that the class of all sets is completely equivalent to the class of all the open sets and the class of all the closed sets. Let and so on, where and each of length is k 3 .We can see that . The Cantor set is the perfect set , we suppose that there exist
By Lemma 2, it is shown as follows:  .Therefore, we obtain from (1) that Finally, the theorem holds.If either the left end of l k I 1 or the right end of r k I 1 is contained inU , let us say that the left end of l k I 1 is contained inU , and the right end of r k I 1 is not contained inU , then there exists a positive integer t as follows: z It is shown as follows: And: We claim that: In fact, it is obvious that (6) holds for It follows that: And: Similar to the above discussion, we can get

2 N
for the set E and the number s

2 m
considered as a self-similarity set generated by two similitudes with the scale factor 3 1 .Since C satisfies the open set condition, the Hausdorff dimension of C equals the similarity dimension of C , be any k th-stage basic interval.We de- note by k F , the class of all the k th-stage basic intervals, and * class of all the basic intervals.By the definition of a net, F is a net for C .Any two basic intervals of F are either disjoint or else one contained in the other.3 SOME RESULTS Since C can be covered by k 2 k th-stage basic in- tervals of length k 3 , and by the definition of the Hausdorff s-dimensional measure, the following lemma holds obviously.Let k I be any k th-stage basic interval, and ^` k m I m ! the class of all the m th-stage basic intervals contained in k I .Then: th-stage basic intervals, each of length is m 3 (countable) covering of C which consists of the basic intervals, and in which each of D cannot completely contain the other.Then: we denote by p m , the number of all the 1 p th-stage basic intervals, which are not con-SHS Web of Conferences 01015-p.2tained in any q th-stage basic intervals in D we have as follows: basic intervals, which is not contained in any q th-stage basic intervals in D where 1 p q .Obviously, p M is non-empty and compact, and

Theorem 2 .F.
of the net measure and Theorem 1, it is obvious that the corollary holds.Let U be an open subset of R , and U be the family of the basic intervals V I contained completely inU , in which no one is contained in the other.Then it is shown as follows: It is obvious that U can only intersect with one k th-stage basic in- terval denoted by I and cannot completely contain

jU 3 ,
basic intervals, next, we prove the theorem by two cases.First, if U intersects with one stage basic interval in I whose left end coin- cides with the left end of I , and by th-stage basic interval in I whose right end coin- cides with the right end of I .Since k U U and U cannot contain both the left end of l k I 1 and the right end of r k I 1 .

I 1 ,
know from (4) and (5) that (3) holds.If U can contain none of both the left end of then there exist two positive integers m and n as follows: z z

F
To prove the opposite inequality, we need to consider all δ-coverings of C by the definition of E H s .Since the class of all sets is completely equivalent to the class of all the open sets, it is enough to prove C coverings of C .Now, let ^ì U be an arbitrary open δ-covering of C .For any given ^ì be the class of the basic intervals contained completely in i U , in which no one is contained in the other, then

Corollary 2 . 1 . 1 ?
For the Cantor set C and the number of all sets is completely equivalent to the net F which is the class consisting of all basic intervals.In[7], the authors pose eight open problems on the exact value of the Hausdorff measure.The first open problem is shown as follows.ProblemUnder what condition is there a covering of E , say Such a covering of E is called as a best covering.It is easy to see that ^C is a best covering of C , and that the class k F of all the k th-stage basic intervals is the best covering of C .By Theorem 1 and Theorem 3, we can obtain the following result.Corollary 3. Let ^ì U D be any covering of C consisting of basic intervals in which each of D cannot completely contain the other, then ^ì U D is the best covering of C .Note that ^ì U D in Corollary 3 may be infinite, therefore, it is non-trivial.
is called a net for E .If for any In this paper, the author further reveals some intrinsic properties of the Cantor set.By erties, the author gives a new method for calculating the exact value of the Hausdorff measure of the C and shows the facts that each covering which consists of basic intervals, in which any basic interval can pletely contain the other, is the best covering of the Cantor set, and the Hausdorff measure of the Cant be determined by coverings which only consist of basic intervals.