Let $G$ be a locally compact group (written additively), $\lambda$ an automophism of $G$, and $\alpha$ a Haar measure in $G$. As the Haar measure is unique up to factor constant, $\lambda$ transform $\alpha$ into $c\alpha$ ($c\in\mathbb R_{>0}$).
My question is:
What does "$\lambda$ transform $\alpha$ into $c\alpha$"?
Thank you all.
The answer to this question, as noted user126033, is no. Indeed, by using the technique of Hamel basis, there is a Lebesgue non-measurable automorhizms $\lambda : R \to R$. So we can not define $\mu$ by the formula $\mu(A)=\alpha(\lambda(A))$, because for some Lebesgue measurable set $A_0$ its image $\lambda(A_0)$ is not Lebesgue measurable.
In the case, when $\lambda$ is Lebesgue measurable, the answer to this question, as noted Cameron Williams, is yes.
P.S. I wanted to do a comment, but I had no 50 reputation.