Home » Mathematics » Equivalent definitions of Hausdorff dimension

Equivalent definitions of Hausdorff dimension

Notations:

  • (X,\Sigma) – Measurable space, (X=\mathbb{R}^n, n\geq 1) where \Sigma denotes the \sigma-algebra
  • We mean measure, to be outer measure, that is function on \{A: A \subset \mathbb{R}^n\} to [0,\infty] such that it is monotone, countably sub-additive and assumes value zero for the empty set.
  • A measure \mu is called Borel measure if the Borel sets (sets formed by open/closed sets) are measurable, that is in \Sigma.
    •  A measure \mu is Borel if and only if for any sets A_1,A_2\subset \mathbb{R}^n with d(A_1,A_2)>0 we have \mu(A_1\cup A_2)=\mu(A_1)+\mu(A_2). (d(A_1,A_2)=\inf\{d(a_1,a_2):a_1\in A_1, a_2\in A_2\} denotes the distance between two sets A_1 and A_2)
  • A Borel measure \mu is Borel regular if for every A\subset \mathbb{R}^n there exists a Borel set B such that A\subset B and \mu(A)=\mu(B).
  • If a Borel measure \mu is not Borel regular, then \tilde{\mu} defined as \tilde{\mu}(A)=\inf\{\mu(B):A\subset B: B- Borel \}, is Borel regular.

 

Equivalent definitions of Hausdorff dimension

 

1. Definition of Hausdorff dimension:

(By Hausdorff via Caratheodory’s construction)

For any set A\subset \mathbb{R}^n we define the \alpha-dimensional Hausdorff measure of A as,

\mathcal{H}^{\alpha}(A)=\lim_{\delta\downarrow 0}\mathcal{H}_{\delta}^{\alpha}(A)

where \mathcal{H}_{\delta}^{\alpha}(A) denotes the infimum of \{c_{\alpha}2^{-\alpha}\sum_{i}d(E_i)^{\alpha}\} taken over all the collection of Borel cover \{E_i\}_i of A with diameter of E_i, d(E_i)\leq \delta.

  • Note that \mathcal{H}_{\delta}^{\alpha}(A)\leq \mathcal{H}_{\epsilon}^{\alpha}(A) for all 0<\epsilon\leq \delta\leq \infty.
  • When \alpha is an integer, c_{\alpha} denotes the volume of the \alpha-dimensional unit ball in \mathbb{R}^n (for 0\leq \alpha\leq n).
  • For convenience, we assume c(\alpha)2^{-\alpha} to be one when we consider \alpha to be non-integer.
  • \mathcal{H}^0 is counting measure.

We observe the following:

  • \mathcal{H}^s is Borel measure
    • It is easy to check that \mathcal{H}_{\delta}^s is monotone, countably sub-additive non-negative measure that assumes value zero for the empty set, that is \mathcal{H}_{\delta}^s is a (outer) measure for all \delta. Hence for any A_1,A_2\subset\mathbb{R}^n,\mathcal{H}_{\delta}^s(A_1\cup A_2)\leq \mathcal{H}_{\delta}^s(A_1)+\mathcal{H}_{\delta}^s(A_2).
    • If A_1,A_2\subset\mathbb{R}^n with d(A_1,A_2)>0, then choose 0<\delta<(1/2)d(A_1,A_2) and by the definition of infimum choose a Borel cover \{E_i\}_i of A_1\cup A_2 with d(E_i), that is diameter of E_i is \leq \delta. Then E_i can either intersect A_1 or A_2, but not both. So we have
      \mathcal{H}_{\delta}(A_1)+\mathcal{H}_{\delta}(A_2)\ \ \ \ \ \ \leq \sum_{A_1\cap E_i \neq \emptyset}d(E_i)^s+\sum_{A_2\cap E_i \neq \emptyset}d(E_i)^s\ \ \ \ \ \ \leq \sum d(E_i)^s

      and thus \mathcal{H}_{\delta}^s(A_1)+\mathcal{H}_{\delta}^s(A_2)\leq \mathcal{H}_{\delta}^s(A_1\cup A_2).

    • That is, \mathcal{H}_{\delta}^s(A_1\cup A_2)= \mathcal{H}_{\delta}^s(A_1)+\mathcal{H}_{\delta}^s(A_2) and hence \mathcal{H}^s(A_1\cup A_2)= \mathcal{H}^s(A_1)+\mathcal{H}^s(A_2). Thus \mathcal{H}^s is Borel measure.
  • \mathcal{H}^s is Borel regular.
    • Fix i. Choose a Borel cover \{E_{ij}\}_j of A such that \sum d(E_{ij})\leq \mathcal{H}_{1/i}^s(A)+1/i. Then B=\cap_i\cup_jE_{ij} is a Borel set containing A with \mathcal{H}^s(B)=\mathcal{H}^s(A).
  • For 0\leq s<t\leq n and A\subset \mathbb{R}^n, \mathcal{H}^s(A)<\infty implies \mathcal{H}^t(A)=0. (In other words, \mathcal{H}^t(A)>0 implies \mathcal{H}^s(A)=\infty)
    • Note that for \delta>0, by the definition of infimum, there exists a Borel cover \{E_i\}_i of A such that \sum d(E_i)^s\leq \mathcal{H}_{\delta}^{s}(A)+1 where d(E_i) denotes the diameter of E_i. Hence\mathcal{H}_{\delta}^t(A)
      \ \ \ \leq \sum_i d(E_i)^t
      \ \ \ \leq \delta^{t-s}\sum_i d(E_i)^s
      \ \ \ \leq \delta^{t-s}(\mathcal{H}_{\delta}^{s}(A)+1)which goes to zero as \delta goes to zero.

 

The Hausdorff dimension of a set A is defined as

dim_{\mathcal{H}}A
\ \ \ =\sup\{s:\mathcal{H}^s(A)>0\}
\ \ \ =\sup\{s:\mathcal{H}^s(A)=\infty\}
\ \ \ =\inf\{s:\mathcal{H}^s(A)=0\}
\ \ \ =\inf\{s:\mathcal{H}^s(A)<\infty\}

We have computed the Hausdorff dimension of generalized Cantor set here.

(to be updated)

This blog is loosely based on the notes prepared while giving a lecture series during January 2017-April 2017 at IISER - Bhopal. The definitions, the notations, proofs, theorems are from the references: 1) W. F. Donoghue; Distributions and Fourier Transforms, Acad. Press, 1969. MR3363413 2) Pertti Mattila, Geometry of sets and measures in Euclidean spaces: Fractals and rectifiability, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995.  MR1333890 3) P. Mattila. Fourier Analysis and Hausdorff Dimension, Cambridge University Press, Cambridge, 2015. 4) Wolff T.H.: Lectures on Harmonic Analysis. University Lecture Series, 29. Amer. Math. Soc., Providence, RI (2003)

 


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