Are Hausdorff measures on the real line Haar measures for some locally compact topology?

Question

For $0\leq d\leq 1$, let $\lambda_d$ be the $d$-dimensional Hausdorff measure on $\mathbb{R}$. Note that it is translation-invariant. Does there exist a locally compact topology $\mathscr{T}_d$ on $\mathbb{R}$, finer than the usual topology and compatible with the (additive) group structure (i.e., $+$ and $-$ are continuous), such that $\lambda_d$ is, up to some normalization, the Haar measure for $(\mathbb{R},+,\mathscr{T}_d)$?

(For $d=1$ the usual topology provides a positive answer. For $d=0$ the discrete topology does. So the question is whether we can do something in between.)

Bonus points if $\mathscr{T}_d$ can somehow be made "canonical".

Answer 1

The answer is NO because the Euclidean and the discrete topologies are the unique locally compact group topologies on $\mathbb R$, which are stronger that the Euclidean topology of the real line.

The reason is that $\mathbb R$ endowed with such topology $\tau$ is a locally compact abelian topological group without small subgroups, so is a Lie group (by the Gleason-Mongomery-Zippin Theorem). Since $(\mathbb R,\tau)$ admits a continuous injective map into $\mathbb R$, it has dimension $\le 1$. If the dimension of the Lie group $(\mathbb R,\tau)$ is 1, then it is (locally) homeomorphic to $\mathbb R$. If the Lie group $(\mathbb R,\tau)$ has dimension zero, then it is discrete.