TITLE: Are Hausdorff measures on the real line Haar measures for some locally compact topology?
QUESTION [13 upvotes]: 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".

REPLY [15 votes]: 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.