Is there a classification of measures on $\mathbb R^n$ which are invariant under (Euclidean) isometries? Hausdorff measures of all kinds are examples -- could that be all of them? More precisely,
By a measure on $\mathbb R^n$, I mean a function $\mu: Borel(\mathbb R^n) \to [0,\infty]$ (where $Borel(\mathbb R^n)$ is the $\sigma$-algebra of Borel subsets of $\mathbb R^n$) such that $\mu$ carries countable disjoint unions to sums (this includes the condition $\mu(\emptyset) = 0$ as the case of the empty disjoint union and nullary sum).
I say that $\mu$ is isometry-invariant if, for every isometry $\Lambda: \mathbb R^n \to \mathbb R^n$, we have $\mu(\Lambda A) = \mu(A)$ for every Borel set $A \subseteq \mathbb R^n$.
Question 1: Is there a classification of isometry-invariant measures on $\mathbb R^n$?
For example, for a suitable "gauge" function $\phi: [0,\infty) \to [0,\infty]$ and a suitable collection of sets $\mathcal C \subseteq Borel(\mathbb R^n)$, the $(\phi,\mathcal C)$-Hausdorff meausure is an isometry-invariant measure on $\mathbb R^n$, defined by
$$\mu^{\phi,\mathcal C}(A) = \lim_{\delta \to 0^+} \inf \{ \sum_i \phi(diam(C_i)) \mid C_i \in \mathcal C,\, A \subseteq \cup C_i,\, diam(C_i) < \delta \}$$
Question 2: In particular, is every isometry-invariant measure on $\mathbb R^n$ given by the $(\phi,\mathcal C)$-Hausdorff measure for some gauge function $\phi$ and collection $\mathcal C$?
NB: When $n=1$, Hirst constructs a $(\phi,\mathcal C)$-Hausdorff measure which is not equivalent to a $(\psi,\mathcal O)$-Hausdorff measure, where $\mathcal O \subseteq Borel(\mathbb R^n)$ is the collection of open sets, for any $\psi$. His $\mathcal C$ is translation invariant, so that his $\mu^{(\phi,\mathcal C)}$ is translation-invariant, but I don't think his $\mathcal C$ is reflection-invariant (so strictly speaking it may no be "suitable" in the sense alluded to above) and as a result, I'm not sure whether his $\mu^{(\phi,\mathcal C)}$ is reflection-invariant. But at any rate, this suggests that in order to construct all isometry-invariant measures via "the Hausdorff method", one will probably need to take advantage of the freedom to vary $\mathcal C$ as well as $\phi$ (if it is indeed possible). I learned of this example from Gro-Tsen's comment here.
Question 3: In light of YCor's excellent comment below, I'd also like to ask both Questions 1 and 2 with the restriction that the measure $\mu$ satisfy some regularity condition (but not so strong a regularity condition as to ensure we have a multiple of Lebesgue measure) and not be $\{0,\infty\}$-valued.
Question 4: What if we assume additionally that our measure $\mu$ satisfies some sort of universal scaling law, i.e. $\mu(\lambda X) = \psi(\lambda \psi^{-1}(\mu(X)))$ for some order-preserving diffeomorphism $\psi: [0,\infty] \to [0,\infty]$? Does this restriction (plus isometry-invariance) make it possible to classify such measures?
