Hausdorffova miera

Definícia: Nech ${\displaystyle \mathbf {A} \subset \mathbb {R} ^{n},0\leq s<\infty ,0<\delta \leq \infty }$ definujeme:

${\displaystyle (i)\mathbf {H} _{\delta }^{s}(A)=\inf\{\sum _{i=1}^{\infty }\alpha (s)({\frac {diam(C_{i})}{2}})^{s}|A\subset {\cup _{j=1}^{\infty }C_{j}},diam(C_{j})\leq \delta \},}$

kde

${\displaystyle \alpha (s)={\frac {\pi ^{\frac {s}{2}}}{\Gamma ({\frac {s}{2}}+1)}}}$

túto

${\displaystyle \Gamma (r)=\int _{0}^{\infty }e^{-x}x^{r-1}dx,(0<r<\infty ),}$

je obyčajná gamma funkcia.

${\displaystyle \mathbf {(} ii)}$Pro ${\displaystyle \mathbf {A} }$ a ${\displaystyle \mathbf {s} }$ s vlastnosťami ako vyššie, definujeme:

${\displaystyle H^{s}(A)=\lim _{\delta \to 0}H_{\delta }^{s}(A)=\sup _{\delta >0}H_{\delta }^{s}(A)}$ ${\displaystyle \mathbf {H} ^{s}}$ nazývame s-dimenzionálnou Hausdorffovou mierou na ${\displaystyle \mathbb {R} ^{n}}$.

${\displaystyle \mathbf {H} ^{s}}$ je Borelova regulárna miera pre ${\displaystyle 0\leq s<\infty }$, nie je ale Radonova miera.

Z toho vyplýva toto:

${\displaystyle (i)\mathbf {H} _{\delta }^{s}}$ je miera.

${\displaystyle (ii)\mathbf {H} ^{s}}$ je miera.

${\displaystyle (iii)\mathbf {H} ^{s}}$ je Borelova miera.

Ďalšie zaujímavé vlastnosti:

${\displaystyle (i)\mathbf {H} ^{0}}$ je čítacia miera.

${\displaystyle (ii)\mathbf {H} ^{1}=\mathbf {L} ^{1}}$ na ${\displaystyle \mathbb {R} ^{n}}$, kde ${\displaystyle \mathbf {L} ^{1}}$ je Lebesgueova miera.

${\displaystyle (iii)\mathbf {H} ^{s}=0}$ na ${\displaystyle \mathbb {R} ^{n}}$ pre všetky ${\displaystyle \mathbf {s>n} }$.

${\displaystyle (iv)\mathbf {H} ^{s}(\lambda A)=\lambda ^{s}\mathbf {H} ^{s}(A)}$ pre všetky ${\displaystyle \lambda >0,A\subset \mathbb {R} ^{n}}$.

${\displaystyle (v)\mathbf {H} ^{s}(L(A))=\mathbf {H} ^{s}(A)}$ pre všetky afinné izometrie ${\displaystyle L:\mathbb {R} ^{n}\to \mathbb {R} ^{n},A\subset \mathbb {R} ^{n}}$.

• Steven G. Krantz: Measure Theory and Fine Properties of Functions, CRC Press LLC, London 2000, ISBN 0-8493-7157-0.