• Source: Frostman lemma
    • In mathematics, and more specifically, in the theory of fractal dimensions, Frostman's lemma provides a convenient tool for estimating the Hausdorff dimension of sets.
      Lemma: Let A be a Borel subset of Rn, and let s > 0. Then the following are equivalent:

      Hs(A) > 0, where Hs denotes the s-dimensional Hausdorff measure.
      There is an (unsigned) Borel measure μ on Rn satisfying μ(A) > 0, and such that




      μ
      (
      B
      (
      x
      ,
      r
      )
      )


      r

      s




      {\displaystyle \mu (B(x,r))\leq r^{s}}


      holds for all x ∈ Rn and r>0.
      Otto Frostman proved this lemma for closed sets A as part of his PhD dissertation at Lund University in 1935. The generalization to Borel sets is more involved, and requires the theory of Suslin sets.
      A useful corollary of Frostman's lemma requires the notions of the s-capacity of a Borel set A ⊂ Rn, which is defined by





      C

      s


      (
      A
      )
      :=
      sup


      {




      (





      A
      ×
      A





      d
      μ
      (
      x
      )

      d
      μ
      (
      y
      )



      |

      x

      y


      |


      s








      )




      1


      :
      μ

      is a Borel measure and

      μ
      (
      A
      )
      =
      1


      }


      .


      {\displaystyle C_{s}(A):=\sup {\Bigl \{}{\Bigl (}\int _{A\times A}{\frac {d\mu (x)\,d\mu (y)}{|x-y|^{s}}}{\Bigr )}^{-1}:\mu {\text{ is a Borel measure and }}\mu (A)=1{\Bigr \}}.}


      (Here, we take inf ∅ = ∞ and 1⁄∞ = 0. As before, the measure



      μ


      {\displaystyle \mu }

      is unsigned.) It follows from Frostman's lemma that for Borel A ⊂ Rn






      d
      i
      m


      H


      (
      A
      )
      =
      sup
      {
      s

      0
      :

      C

      s


      (
      A
      )
      >
      0
      }
      .


      {\displaystyle \mathrm {dim} _{H}(A)=\sup\{s\geq 0:C_{s}(A)>0\}.}



      Web pages


      Illustrating Frostman measures


      Further reading


      Mattila, Pertti (1995), Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, ISBN 978-0-521-65595-8, MR 1333890

    Kata Kunci Pencarian: