Let $G$ be a locally compact group, and let $\mu$ be a radon measure. I wanna show that the The measure $\mu$ is a left Haar measure on $G$ if and only if the measure $\tilde{\mu}$, defined by $\tilde{\mu}(A) = \mu(A^{−1})$ for $A \in \mathcal B$, is a right Haar measure on $G$.
We can observe the following: $$\tilde{\mu}(gA)=\mu((gA)^{-1})=\mu(A^{-1} g^{-1} )$$ Furthermore we have that $g \mapsto g^{-1}$ is a bijection but how can I now conclude? Many thanks for some help!
EDIT: I found this one:
Does the last if and only if holds because $g\mapsto g^{-1}$ is a bijection?