My question is
Why is the Haar measure on a Lie Group unique upto scalar multiple?
I know how to show it for $\mathbb R^n$ because there I have countable many open balls that form a base and the measure of the unit ball around 0 gives the constant scalar.
How to show for general Lie Group?
Let $\mu,\nu$ be two left Haar measures on a locally compact group. Then $\mu+\nu$ is a left Haar measure, and $\mu$ has a density $f$ with respect to it. Then $f$ is left-invariant, so is constant. Hence $\mu$ is a scalar multiple of $\mu+\nu$.
I don't really see how to make it simpler, even with additional assumptions?