# Is there a nontrivial left-invariant Borel measure (necessarily not Radon) that is not a constant multiple of Haar?

I know that the Haar measure on a locally compact second-countable group $G$ is unique up to a constant multiple.

That is, any nontrivial left-invariant Radon measure is a constant multiple of the Haar measure.

Equivalently (since second-countable), any nontrivial left-invariant locally finite (finite on compact sets) measure is a constant multiple of the Haar measure.

Is it possible that there is a nontrivial left-invariant Borel measure (not locally finite) that is not a constant multiple of the Haar measure?

• Take a sequence $A_n \supset A_{n+1}$ of neighborhoods of $g=e$ such that $\bigcap_n A_n = \{e\}$ and $\mu(A_n) \ne 0$, consider the operator $A_n \ast E= \{g \in G, \exists h \in A_n, hg \in E\}$. I think your counting measure is $\nu(E) = \lim_{n \to \infty} \displaystyle\frac{\mu(A_n \ast E)}{\mu(A_n)}$, while the Haar measure satisfies $\mu(E) = \lim_{n \to \infty} \mu(A_n \ast E)$ – reuns Nov 30 '16 at 11:29