HausdorffTypeMeasureandDimensionPrint资源-CSDN文库

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根据给定文件的内容，我们可以提取以下知识点： 1. Hausdorff测度概念：在数学中，Hausdorff测度是描述几何对象复杂性的一种方式，它可以量度集合的“大小”而不仅限于一维线段、二维平面或三维体积。Hausdorff测度是用于度量Fractal集合或复杂几何结构的强有力的工具。 2. Hausdorff维数：Hausdorff维数是一种广义维数的概念，它是度量空间内子集维度的一种扩展。与传统的拓扑维数相比，Hausdorff维数可以为分形等非整数维数的对象提供一个精确的测量值。 3. Lebesgue测度：Lebesgue测度是数学中定义在欧几里得空间的一个测度理论，它为可测集提供了一种“长度”、“面积”、“体积”等概念的准确度量。Lebesgue测度是对经典测度理论的重要扩展，其扩展了对于可测集的概念，能够处理更复杂的几何结构。 4. 平面中新的Hausdorff型测度：文章中介绍了在平面上引进的一种新的Hausdorff型测度，这种测度为平面内任意非空子集F定义了一个与覆盖直径有关的测度。通过集合覆盖的方式，可以逼近一个极限，这个极限定义为集合F的Hausdorff测度。随着覆盖直径δ的减小，测度增加，并且趋向一个极限值。 5. Hausdorff维数的定义：利用Hausdorff测度，可以定义Hausdorff维数。集合F的Hausdorff维数被定义为使得Hausdorff测度为零的最小维度s，以及使得Hausdorff测度为无穷大的最大维度s。Hausdorff维数为这两个极限的交集。 6. 三角形覆盖与新维度打印：文章中定义了利用三角形来覆盖集合F并定义了相关的测度。对于平面内的一个三角形，定义了边长b、c和a，通过这些边长的幂函数定义了一个新的测度。三角形的四个顶点被标记为p1、p2、p3和p4，而对应的边长的幂函数是这个测度的基础。 7. 测度函数H的性质：新定义的测度函数H被证明是外测度。在数学中，外测度是一种定义在集合的幂集上的函数，它满足一系列公理，包括非负性、零测度的集合的测度为零，以及可数可加性。 8. 主要定理的含义：文档提到的主要定理可能阐述了这种新的Hausdorff型测度是Lebesgue测度在平面上的一个推广形式，同时新的维数打印（维度表示）包含了比传统维数更多的信息。 9. 研究的潜在意义：通过Hausdorff型测度和维数打印在数学中的应用，可以在分析几何结构的复杂性、理解分形结构以及在理论数学之外的应用，例如图像分析、数据压缩和分形艺术创作中具有潜在应用价值。 10. 数学关键词：文章中提到的关键词包括“Hausdorff型测度”、“维数打印”、“Lebesgue测度”，这些是研究中重要的专业术语，它们涉及了现代数学的前沿研究领域，特别是在测度论和分形几何方面。文件介绍了一篇关于Hausdorff型测度和维度打印的研究论文。论文的主要贡献是提出了一个新的测度系统，该系统基于平面内的集合覆盖，并通过引入三角形覆盖来定义新的维数打印。这一研究不仅推广了经典的Lebesgue测度，而且还增加了对分形等复杂结构理解的维度信息。

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-1-

HAUSDORFF TYPE MEASURE AND

DIMENSION PRINT

Han-Ju Li and Xuan-Ming Mei

School of Mathematical Sciences

South China Normal University

Guangzhou, P.R. C 510631

hanjuli@yahoo.com.cn

Abstract

We introduce a new Hausdorff type measure on plane and define a new dimension

print associated with this measure. Our main theorems show that this measure is

just the generalization of Lebesgue measure on plane and the new dimension

print contains more information than the usual one.

Keywords: Dimension print, Hausdorff type measure, Lebesgue measure

2000 Mathematics Subject Classification. 28A78

1 Introduction

Let

U be any nonempty subset of

 , the diameter ofU is defined as| | sup{| |: , }UxyxyU

−∈.

{}

∞

is a countable (or finite) collection of sets of diameter at most

that cover F , i.e.,

∞

⊂

∪

and

0< U

≤

for every

i , then we say that{}

U is a

-cover of

. Suppose that

F ⊂  and s is a non-negative

number. For any

> , we define

( ) inf :{ } is a -cover of

HF U U F

∞

⎧

⎫

⎨

⎬

⎩⎭

∑

Clearly, as

decreases, ()

increases and so ()

approaches a limit as 0

→ . We write

() lim ()

HF HF

→

We call

()

the s -dimension Hausdorff measure of

. It is well known that the set function

H is

a metric outer measure on

 . The Hausdorff dimension of

F ⊂  is defined as

{

}

{

}

dim inf 0 : ( ) 0 sup 0 : ( )

FsHF sHF=≥ ==≥ =∞

Now we define a new Hausdorff type measure on

 . Let

be a triangle in plane and let ()a Δ ,

()b Δ and ()c Δ be the length of the sides of

. We assume that () () ()abc

≥Δ≥Δ. Denote

() () ()

():

abc

Δ+ Δ+ Δ

Δ=

() () ()

():

abc

+Δ−Δ

Δ=

() () ()

():

acb

Δ+ Δ− Δ

Δ=

，

() () ()

():

bca

+Δ−Δ

Δ=

Clearly,

1234

() () () () 0ppppΔ≥ Δ≥ Δ≥ Δ> .

-2-

Denote

1234 1234

:{(, , , ): , , , 0}ssss ssss

=≥ . For

F ⊂  and

1234

(, , , )sssss

=∈



 , let

( ) inf ( ) :{ } is a -cover of by triangles

Fp F

∞

⎧⎫

=ΔΔ

⎨⎬

⎩⎭

∑



where

12 4

12 3 4

(): () () () ()

iiiii

pppppΔ= Δ Δ Δ Δ .

Clearly, as

decreases, ()



increases and so ()



approaches a limit as

→

, we write

() lim ()

→



We can prove that the set function



is a metric outer measure. So the measure



is a Borel

measure on plane.

In 1919, Hausdorff introduced [1] a concept of Hausdorff dimension which can distinct two null Lebe-

sgue sets. For example, let



denote the rational numbers and C denote the Cantor set. Both of them

are Lebesgue null set, while

dim 0

 and

log 2

dim

log 3

C = .

In 1988, Rogers [2] introduced a concept of Dimension print, which contains more detailed information

than the Hausdorff dimension. For instance, Dust-like set formed by the product of two uniform Cantor

sets of Hausdorff dimension

34 and Stratified set formed by the product of a uniform Cantor set of

Hausdorff dimension

and a line segment have the same Hausdorff dimension

but with different

dimension prints. Dimension print was introduced in [5] and was further studied by [3].

Now we give the definition of dimension print due to Rogers as follows. Let

U be a rectangle on plane

and

() ()aU bU≥ be the length of the sides of

. Let

and t be the non-negative numbers. For any

subset

F of

 , let

(,)

( ) inf ( ) ( ) :{ } is a -cover of by rectangles

st s t

iii

FaUbUU F

∞

⎧⎫

⎨⎬

⎩⎭

∑

By letting

→ , we get a Hausdorff type measure

(,)

 , we write

(,) (,)

() lim ()

st st

→

The dimension print associated with

(,)

is defined as

{

}

(,)

print( ) : ( , ): 0, 0, ( ) 0

Fstst F=≥≥ >

The following properties are well known for us.

(1) If

FF⊂ , then

print( ) print( )FF⊂ .

(2)

print ( ) (print )

ii i

∪∪ .

(3) If

(,) printst F∈ , and

,stsatisfies

st st

≤+and

≤

, then

(,) print()st F∈ .

In this paper, we introduce a new dimension print associated with



as follows. For a subset F of

 ,

let

{

}

print ( ) : : , ( ) 0

Fss F

∗

∈>





. (1.1)

We will study the relations between

print ( )F

∗

and print( )F in section 3. Several comments are

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