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 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)