$\newcommand{\RR}{\mathbb{R}}\newcommand{\calF}{\mathcal{F}}\newcommand{\diam}{\mathrm{diam}}$ 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".