If $dx$ is a Haar measure on $\mathbb{Q}_p$, it is said that $d^\times x = |x|_p^{-1} dx$ is a Haar measure for $\mathbb{Q}_p^\times$, or in other words that $d(ax) = |a|_p dx$. How does one determine such relations.
Only using the definition of the Haar measure as invariant measure I do not make sense of how to prove such properties.
Another instance of this is that, when considering the parabolic subgroup $$\pmatrix{a&b\\& c}$$
the left Haar measure is apparently $\frac{da db dc}{|ac|}$ while rhe right Haar measure is $\frac{dadbdc}{|a|^2}$. Even knowing the explicit formulas, I don't understand how to verify they are indeed Haar measures.