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Hausdorffova míra (dále
H
s
{\displaystyle \mathbf {H} ^{s}}
míra na
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
Felix Hausdorff . Základní myšlenkou je, že množina
A
{\displaystyle \mathbf {A} }
R
n
{\displaystyle \mathbb {R} ^{n}}
0
<
H
s
(
A
)
<
∞
{\displaystyle 0<H^{s}(A)<\infty }
i když
A
{\displaystyle \mathbf {A} }
H
s
{\displaystyle \mathbf {H} ^{s}}
Definice : Nechť
A
⊂
R
n
,
0
≤
s
<
∞
,
0
<
δ
≤
∞
{\displaystyle \mathbf {A} \subset \mathbb {R} ^{n},0\leq s<\infty ,0<\delta \leq \infty }
(
i
)
H
δ
s
(
A
)
=
inf
{
∑
i
=
1
∞
α
(
s
)
(
d
i
a
m
(
C
i
)
2
)
s
|
A
⊂
⋃
j
=
1
∞
C
j
,
d
i
a
m
(
C
j
)
≤
δ
}
,
{\displaystyle (i)\,\mathbf {H} _{\delta }^{s}(A)=\inf \left\{{\sum _{i=1}^{\infty }\alpha (s)\left({\frac {\mathrm {diam} (C_{i})}{2}}\right)^{\!\!s}}\;{\Big |}\,{A\subset {\bigcup _{j=1}^{\infty }C_{j}},\mathrm {diam} (C_{j})\leq \delta }\right\},}
α
(
s
)
=
π
s
2
Γ
(
s
2
+
1
)
{\displaystyle \alpha (s)={\frac {\pi ^{\frac {s}{2}}}{\Gamma ({\frac {s}{2}}+1)}}}
Γ
(
r
)
=
∫
0
∞
e
−
x
x
r
−
1
d
x
,
(
0
<
r
<
∞
)
,
{\displaystyle \Gamma (r)=\int _{0}^{\infty }e^{-x}x^{r-1}dx,(0<r<\infty ),}
gamma funkce .
(
i
i
)
{\displaystyle \mathbf {(} ii)}
A
{\displaystyle \mathbf {A} }
s
{\displaystyle \mathbf {s} }
H
s
(
A
)
=
lim
δ
→
0
H
δ
s
(
A
)
=
sup
δ
>
0
H
δ
s
(
A
)
{\displaystyle H^{s}(A)=\lim _{\delta \to 0}H_{\delta }^{s}(A)=\sup _{\delta >0}H_{\delta }^{s}(A)}
H
s
{\displaystyle \mathbf {H} ^{s}}
R
n
{\displaystyle \mathbb {R} ^{n}}
Elementární vlastnosti Hausdorffovy míry [ editovat | editovat zdroj ]
H
s
{\displaystyle \mathbf {H} ^{s}}
0
≤
s
<
∞
{\displaystyle 0\leq s<\infty }
(
i
)
H
δ
s
{\displaystyle (i)\mathbf {H} _{\delta }^{s}}
míra .
(
i
i
)
H
s
{\displaystyle (ii)\mathbf {H} ^{s}}
(
i
i
i
)
H
s
{\displaystyle (iii)\mathbf {H} ^{s}}
borelovská míra .
(
i
)
H
0
{\displaystyle (i)\mathbf {H} ^{0}}
čítací míra .
(
i
i
)
H
1
=
L
1
{\displaystyle (ii)\mathbf {H} ^{1}=\mathbf {L} ^{1}}
R
n
{\displaystyle \mathbb {R} ^{n}}
L
1
{\displaystyle \mathbf {L} ^{1}}
Lebesgueova míra .
(
i
i
i
)
H
s
=
0
{\displaystyle (iii)\mathbf {H} ^{s}=0}
R
n
{\displaystyle \mathbb {R} ^{n}}
s
>
n
{\displaystyle \mathbf {s>n} }
(
i
v
)
H
s
(
λ
A
)
=
λ
s
H
s
(
A
)
{\displaystyle (iv)\mathbf {H} ^{s}(\lambda A)=\lambda ^{s}\mathbf {H} ^{s}(A)}
λ
>
0
,
A
⊂
R
n
{\displaystyle \lambda >0,A\subset \mathbb {R} ^{n}}
(
v
)
H
s
(
L
(
A
)
)
=
H
s
(
A
)
{\displaystyle (v)\mathbf {H} ^{s}(L(A))=\mathbf {H} ^{s}(A)}
L
:
R
n
→
R
n
,
A
⊂
R
n
{\displaystyle L:\mathbb {R} ^{n}\to \mathbb {R} ^{n},A\subset \mathbb {R} ^{n}}
Steven G. Krantz: Measure Theory and Fine Properties of Functions,
CRC Press LLC, London 2000, ISBN 0-8493-7157-0 
