Geometric measures different from Hausdorff

Question

$\newcommand{\RR}{\mathbb{R}}\newcommand{\calF}{\mathcal{F}}\newcommand{\diam}{\mathrm{diam}}$ In geometric measure theory there are various notions of $m$-dimensional measure for sets $A\subset \RR^n$ for $m\leq n$ (some of them also for non-integer $m$, but this is not the point here). They all build on Carathéodory's general construction which works by covering $A$ with countably many sets $E_i$ from a specific base family $\calF$, measuring the sets $E_i$ with a function $\zeta$ and then building the infimum over all these coverings that are $\delta$-fine (i.e. $\zeta(E_i)\leq \delta$), and letting $\delta\to 0$.

Among these specific measures are

• Hausdorff measure for $s>0$: This uses all sets for covering and takes $\zeta = \diam^s$, the diameter of the set raised to the $s$th power.
• Spherical measure is the same but restricts the family $\calF$ to consist only of balls.
• Dyadic net measure is again similar but uses cubes with dyadic corner points as the family $\calF$.
• Gross measure for $m=0,1,\dots$ is a bit different: It uses the Borel sets for $\calF$ and reuses the $m$-dimensional Lebesgue measure as follows. For some $E$ define $\zeta(E)$ as the largest $m$-dimensional Lebesgue measure you get by projecting $E$ onto any $m$-dimensional subspace.
• Carathéodory measure is similar to Gross measure but takes as $\calF$ only closed and convex subsets.

(There are more, e.g. Federer measure or Gillespie measure…)

Somehow, the Gross measure seems most natural to me as the method of covering really drives the minimizing covers to follow the set as close as possible and also directly counts the size of the covering sets in the way one wants to have in the end (slight drawback is that it only works for integer $m$). However, the Hausdorff measure seems to produce a very reasonable definition as illustrated by various examples (e.g. rectifiable curves, Cantor dust in two dimensions with Hausdorff dimension 1).

In the books on geometric measure theory I considered (like Federer, Mattila, Krantz & Parks, Morgan) they describe the construction and basic estimates between them quite detailed and mainly use the Hausdorff measure afterwards. Especially for the notion of dimension it turns out that Hausdorff measure, spherical measure and the dyadic net measure all give the same notion of dimension.

However, I could not find answers to these questions:

1. Are there sets for which the Gross measure gives something unreasonable (in comparison to intuition or Hausdorff measure), e.g. a totally "wrong" size or even a "wrong" dimension?

2. Are there sets for which the spherical or dyadic net measure gives some intuitively wrong size (or some size different from the Hausdorff measure)?

I would also love to have some examples of the usefulness of the measures different from the Hausdorff measure (which appear to be handy for the analysis of self-similar sets). Hence, a third question is

${}$3. What are the spherical, dyadic net, Gross or Carathéodory measure good for?

Somehow I am most interested in Gross and Carathéodory measure - they are defined in several books but basically not used. Also it is daunting to search the web for "Gross measure" and totally not helpful to search for "Carathéodory measure".

Answer 1

Hausdorff, spherical Hausdorff and dyadic net measures not only give rise to the same dimension but, for a fixed value of $m$, are comparable up to constants that depend only on the ambient dimension $d$. In particular, the property of having zero, positive and finite, or infinite measure coincides for these three measures.

I am sure that there are examples (even fairly simple examples such as self-similar sets) for which the actual values of these three measures differ, but I don't have a concrete example or reference at hand.

As you say Hausdorff dimension is the most used, and the reason to consider spherical or dyadic net measures is that they sometimes make calculations simpler (while yielding the same notion of dimension, and even of zero/infinite measure).

One very useful way to study the size of sets is by representing them as trees, and a universal way to do so in Euclidean space is via dyadic cubes - when doing this dyadic net measures are more natural. For example, the standard proof of Frostman's lemma (one the most basic results about Hausdorff measures) uses dyadic partitions and dyadic net measures, although it is often stated for Hausdorff measure.

On the other hand, covering by spheres are often easier to analyze than covering by arbitrary sets, and for this reason spherical measures is sometimes easier to compute (if one is seeking the exact value). In particular, the study of densities (the behavior of $\mu(A\cap B(x,r))/r^m$ as $r\to 0$, where $\mu$ is some measure of interest) is easier for spherical measures

For nice regular sets (such as embedded manifolds) all three measures do agree, so I don't think there are examples where size according to spherical or net measures is "totally wrong".

I hadn't heard of Gross or Caratheodory measures before, but here are some general remarks. I'll denote Gross measure by $G_m$, Hausdorff measure by $H_s$ and Lebesgue measure by $L_m$.

Since they are defined only for integer $m$, it does not make too much sense to consider a dimension associated to Gross measure, but in some sense it agrees with the notion of Hausdorff dimension. If $\dim_H A>m$ (where $A\subset\mathbb{R}^n$ and $\dim_H$ is Hausdorff dimension), then $G_m(A)>0$. This follows from the Marstrand-Mattila projection theorem, that says that if $\dim_H A>m$, then the projection of $A$ onto almost every $m$-dimensional subspace has positive $L_m$-measure. In fact it is enough to know this for only one projection. On the other hand, $G_m$ is bounded above by a constant multiple of $H_m$ (since projecting does not increase Hausdorff measure, and on an $m$-dimensional subspace, $H_m$ is a constant multiple of Lebesgue measure). In particular, if $\dim_H(A)<m$, then $H_m(A)=0$ and so $G_m(A)=0$.

However, it is possible that $H_m(A)>0$ but $G_m(A)=0$. Indeed, there exist sets of positive and finite $m$-dimensional measure such that their projection onto any $m$-dimensional subspace has zero $L_m$-measure. For example, this is the case if $A$ is a self-similar set satisfying a suitable separation condition if the orthogonal parts of the generating similitudes generate a dense subgroup of the orthogonal group (this was proved by Eroglu in the plane and Farkas in arbitrary dimension, but ad hoc examples were known long before). For my intuition, it is "correct" to say that such self-similar sets have positive $m$-measure, so Gross measure looks "wrong" to me here - but this is likely simply because I'm much more familiar with Hausdorff measure.

Everything I've said about Gross measure applies equally well to Carathéodory measure.