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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
n
, the diameter ofU is defined as| | sup{| |: , }UxyxyU
=
−∈.
If
1
{}
ii
U
∞
=
is a countable (or finite) collection of sets of diameter at most
δ
that cover F , i.e.,
1
i
i
FU
∞
=
⊂
∪
and
i
0< U
δ
≤
for every
i , then we say that{}
i
U is a
δ
-cover of
F
. Suppose that
n
F ⊂ and s is a non-negative
number. For any
0
δ
> , we define
1
( ) inf :{ } is a -cover of
s
s
ii
i
HF U U F
δ
δ
∞
=
⎧
⎫
=
⎨
⎬
⎩⎭
∑
.
Clearly, as
δ
decreases, ()
s
HF
δ
increases and so ()
s
HF
δ
approaches a limit as 0
δ
→ . We write
0
() lim ()
ss
HF HF
δ
δ
→
=
.
We call
()
s
H
F
the s -dimension Hausdorff measure of
F
. It is well known that the set function
s
H is
a metric outer measure on
n
. The Hausdorff dimension of
n
F ⊂ is defined as
{
}
{
}
dim inf 0 : ( ) 0 sup 0 : ( )
ss
H
FsHF sHF=≥ ==≥ =∞
.
Now we define a new Hausdorff type measure on
2
. Let
Δ
be a triangle in plane and let ()a Δ ,
()b Δ and ()c Δ be the length of the sides of
Δ
. We assume that () () ()abc
Δ
≥Δ≥Δ. Denote
1
() () ()
():
2
abc
p
Δ+ Δ+ Δ
Δ=
,
2
() () ()
():
2
abc
p
Δ
+Δ−Δ
Δ=
,
3
() () ()
():
2
acb
p
Δ+ Δ− Δ
Δ=
，
4
() () ()
():
2
bca
p
Δ
+Δ−Δ
Δ=
.
Clearly,
1234
() () () () 0ppppΔ≥ Δ≥ Δ≥ Δ> .
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