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".


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.


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.