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

Question

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?

Answer 1

Sure. For example, counting measure is trivially invariant and if the group is non-discrete, it is not a constant multiple (or even absolutely continuous with respect to) of the Haar measure.