Perhaps this is a silly question with a "simple" counterexample, but I am not an expert in measure theory, so I thought of asking here.
Basically, I am curious about uniqueness of $\sigma$-finite left-invariant measures on measurable groups. I do not care about existence.
More specifically, let $(G,\Sigma)$ be a measurable group. Suppose $\mu$ is a $\sigma$-finite measure on $(G,\Sigma)$ which is also left-invariant. Is $\mu$ unique in the family of $\sigma$-finite measures? If not, are there "well-known" counterexamples?