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

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".

1 Answer

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.