Metric dimensions properties not in $\mathbb{R}^d$

Question

I have been looking for references on Hausdorff and Box-counting dimensions not in subsets of $\mathbb{R}^d$. The definitions for these terms do not depend on being in $\mathbb{R}^d$, but mostly I have found the discussion only in $\mathbb{R}^d$. However, when I look at properties I do not see why we need to be in $\mathbb{R}^d$, and thought to ask whether the following properties are still true:

1. If $(X,d)$ and $(Y,\rho)$ are metric spaces and $f:X\to Y$ is an $\alpha$-Holder function, then $\dim\big( f[A] \big) \leq \frac{1}{\alpha} \dim(A)$.
2. If $(X,d)$ and $(Y,\rho)$ are metric spaces and $f:X\to Y$ is a bi-Lipschitz function, then $\dim(A)=\dim \big( f[A] \big) $.
3. The dimension can be determined by covers only consisting of balls instead of sets with a condition on their diameter.
4. Does Frostman's lemma also hold in proper metric spaces?

The proofs of the first three properties do not see, rely upon us being in $\mathbb{R}^d$. The fourth is just a hopeful wish that if one borrows nice enough properties from $\mathbb{R}^d$, then the lemma should also hold.

I am trying to compute a Hausdorff dimension of a set in $\mathbb{R}$ which is an image under a Lipschitz map from a metric space not induced from $\mathbb{R}^d$, so I was trying to check what tools do indeed transfer to the general case. I would also appreciate a reference dealing with this general case, if someone knows of one.

Answer 1

Since no one else has answered (or even commented) after my initial comment in which I cited three possibly relevant Ph.D. dissertations (all freely available online), what follows is an expanded version of that comment. The references below should at least provide a starting point for what you want.

Edgar's book [1] is probably the first place I'd look, since he includes a lot of traditional-type measure-theoretic issues, much of it in a metric space setting. Regarding general gauge functions (i.e. finer measurements of “dimension” than provided by power functions), McClure [4] and Rogers [5] are recommended, and see also the Mathematics Stack Exchange question Existence of dimension function (i.e., exact gauge function).

[1] Gerald Arthur Edgar, Integral, Probability, and Fractal Measures, Springer-Verlag, 1998, x + 286 pages. [Bull. AMS review—primarily an extensive overview of the subject]

[2] John David Howroyd, On the Theory of Hausdorff Measures in Metric Spaces, Ph.D. Dissertation (under David Preiss), University College London, June 1994, iii + 69 pages.

Abstract (first 4 paragraphs)

In this work the main objective is to extend the theory of Hausdorff measures in general metric spaces. Throughout the thesis Hausdorff measures are defined using premeasures. A condition on premeasures of ‘finite order’ is introduced which enables the use of a Vitali type covering theorem. Weighted Hausdorff measures are shown to be an important tool when working with Hausdorff measures defined by a premeasure of finite order.

The main result of this thesis is the existence of subsets of finite positive Hausdorff measure for compact metric spaces when the Hausdorff measure has been generated by a premeasure of finite order. This result then extends to analytic subsets of complete separable metric spaces by standard techniques in the case when the increasing sets lemma holds. The proof of this result uses techniques from functional analysis. In this respect the proof presented is quite different from those of the previous literature.

A discussion on Hausdorff–Besicovitch dimension is also to be found. In particular the problem of whether $$\dim (X) + \dim (Y) \; \leq \; \dim (X \times Y) $$ is solved in complete generality. Generalised dimensions involving partitions of Hausdorff functions are also discussed for product spaces. These results follow from a study of the weighted Hausdorff measure on product spaces.

An investigation is made of the sufficiency of some conditions for the increasing sets lemma to hold. Some counterexamples are given to show insufficiency of some of these conditions. The problem of finding a counterexample to the increasing sets lemma for Hausdorff measures generated by Hausdorff functions is also examined. It is also proved that for compact metric spaces we may also approximate the weighted Hausdorff measure by finite Borel measures that are ‘dominated’ by the premeasure generating the weighted Hausdorff measure.

[3] Helen Janeith Joyce, Packing Measures, Packing Dimensions, and the Existence of Sets of Positive Finite Measure, Ph.D. Dissertation (under David Preiss), University College London, September 1995, 100 pages. Abstract

A number of definitions of packing measures have been proposed at one time or another, differing from each other both in the notion of packing they employ, and in whether the radii or the diameters of the balls of the packing are used. In Chapter 1 various definitions of packing measures are considered and relationships between these definitions established.

Chapter 2 presents work which was done jointly with Professor D. Preiss, and which has been published as such. It is shown here that, with one of the possible radius-based definitions of packing measure, every analytic metric space of infinite packing measure contains a compact subset of positive finite measure. It is also indicated how this result carries over to other radius-based packing measures in the case of Hausdorff functions satisfying a doubling condition.

In Chapter 3 a construction is described which provides, for every Hausdorff function $h,$ a compact metric space of infinite diameter-based $h$-packing measure, with no subsets of positive finite measure. It is then indicated how such a construction may be modified to deal with certain Hausdorff functions which do not satisfy a doubling condition, and a radius-based definition of packing measure. In Chapter 4 we consider topological and packing dimensions, and show that if $X$ is a separable metric space, then $$ \dim_{\mathcal T}(X) \; = \; \min\{\dim_{\mathcal Q}(X'): \; X' \; \text{is homeomorphic to} \; X \}, $$ where $\dim_{\mathcal Q}$ denotes the packing dimension associated with any one of the packing measures considered in this work, and $\dim_{\mathcal T}$ denotes topological dimension.

Chapter 5 answers the question, for which Hausdorff functions $h$ may the Hausdorff and packing measures, ${\mathcal H}^h |_A$ and ${\mathcal P}^h |_A,$ agree and be positive and finite for some $A \subseteq {\boldsymbol R}^n.$ We show that the assumption that the two measures agree and are positive and finite on some subset of ${\boldsymbol R}^n$ implies that the function $h$ is a regular density function (in the sense of Preiss). The converse result is also provided, that for each regular density function $h,$ there is a subset $A$ of ${\boldsymbol R}^n$ such that ${\mathcal H}^h |_A = {\mathcal P}^h |_A$ and this common measure is positive and finite.

[4] Mark Christian McClure, Fractal Measures on Infinite Dimensional Sets, Ph.D. Dissertation (under Gerald Arthur Edgar), Ohio State University, 1994, v + 66 pages.

Abstract [from Dissertation Abstracts International 55 #6 (December 1994), p. 2239−B]

This dissertation deals with sets $E$ which are infinite dimensional in the sense that ${\mathcal H}^n (E) = \infty$ for every $n >0.$ Here, ${\mathcal H}^n$ denotes the $n$-dimensional Hausdorff measure. Of course, ${\mathcal H}^n$ may be generalized to ${\mathcal H}^{\varphi},$ where $\varphi (\varepsilon)$ is a more arbitrary function than ${\varepsilon}^n.$ If $E$ is $\sigma$-totally bounded, then ${\mathcal H}^{\varphi}(E) = 0$ for some $\varphi$ and so these generalized measures allow some degree of dimensional analysis.

The first type of example is the hyperspace. Given a metric space $(X,\rho),$ a hyperspace corresponding to $X$ is a metric space whose elements are subsets of $X.$ For example, if ${\mathcal K}(X)$ denotes the set of compact subsets of $X$ and $\tilde{\rho}$ is the Hausdorff metric on ${\mathcal K}(X),$ then $({\mathcal K}(X),\,\tilde{\rho})$ is a hyperspace. Furthermore, $\dim({\mathcal K}(X))$ is related to $\dim (X).$ In a nice situation, one might hope that $$ \dim (X) \sim \varphi \; \implies \; \dim ({\mathcal K}(X)) \sim 2^{-1/\varphi}.$$ So, for example, $\dim({\mathcal K}({\boldsymbol R}^n)) \sim 2^{-(1/e)^n}.$ I give theorems and counterexamples investigating to what extent this statement is true using various interpretations of $\sim$ and definitions of dimension. Analysis is also done on ${\mathcal C}({\boldsymbol R}^n),$ the set of convex subsets of ${\boldsymbol R}^n.$

The second major example is function space. Kolmogorov and his school initiated this study in the 1950’s. Given a set $\mathcal F$ of functions precompact with respect to the uniform norm, they found bounds on $\dim ({\mathcal F})$ in terms of the dimension of the domain of $f \in \mathcal F$ and the degree of differentiability of $f \in \mathcal F.$ Their measure of dimension was what is now called the entropy index. I extend this analysis to some other notions of dimension, including the Hausdorff dimension.

[5] Claude Ambrose Rogers, Hausdorff Measures, Cambridge University Press, 1970, viii + 179 pages. [2nd edition, Cambridge Mathematical Library, Cambridge University Press, 1998, xxx + 195 pages]

The 1998 2nd edition is a reprinting of the 1970 edition that includes “$\ldots$ making many small corrections, a few larger ones, a number of minor additions at the chapter ends, $\ldots$” (from Note on the second edition). The 1998 edition also includes the following: 21 page Foreword (pp. vii−xxvii; 94 references) by Kenneth John Falconer that gives a brief survey of some of the developments since the 1970 edition; 16 page Appendix A: Dimension Prints (by Rogers; pp. 177−192; 4 references).

[6] Richard Curtis Willmott, Hausdorff Measures in Topological Spaces, Ph.D. Dissertation (under Maurice Sion), University of British Columbia, June 1965, vii + 119 pages.

Perhaps a bit more abstract and general than you want, but included in case it's of interest. See also: Sion/Willmott, Hausdorff measures on abstract spaces, Transactions of the American Mathematical Society 123 # 2 (June 1966), pp. 275−309; Willmott, Some properties of Hausdorff measures on uniform spaces, Proceedings of the London Mathematical Society (3) 17 #3 (July 1967), pp. 513−529.