The following is extracted from 'Fourier Analysis on Groups' by Rudin, page $2.$
If $m$ and $m^{\prime}$ are two Haar measures on $G,$ then $m^{\prime} = \lambda m,$ where $\lambda$ is a positive constant.
Proof:
Fix $g \in C_c(G)$ so that $$\int_G g dm = 1.$$ Define $\lambda$ by $$\int_G g(-x)dm^{\prime}(x) = \lambda.$$ For any $f \in C_c(G),$ we then have \begin{align*} \int_G fdm^{\prime} & = \int_G g(y)dm(y) \int_G f(x) dm^{\prime}(x) \\ & = \int_G g(y)dm(y) \int_G f(x+y) dm^{\prime}(x) \\ & = \int_G dm^{\prime}(x) \int_G g(y) f(x+y) dm(y) \\ & = \int_G dm^{\prime}(x) \int_G g(y-x) f(y) dm(y) \\ & = \int_G f(y) dm(y) \int_G g(y-x)dm^{\prime}(x) \\ & = \lambda \int_G f dm. \end{align*} Hence, $m^{\prime} = \lambda m.$ Note that the use of Fubini's theorem was legitimate in the preceding calculation, since the integrands $g(y) f(x+y)$ and $g(y-x)f(y)$ are in $C_c(G \times G).$
Questions: $(1)$ Why the following equality holds $$\int_G g(y)dm(y) \int_G f(x+y) dm^{\prime}(x) = \int_G dm^{\prime}(x) \int_G g(y) f(x+y) dm(y)?$$ Since $f$ contains $y,$ I thought we cannot 'move' $f$ to left integral?
$(2)$ How to use Fubini's theorem to obtain $$ \int_G dm^{\prime}(x) \int_G g(y) f(x+y) dm(y)= \int_G dm^{\prime}(x) \int_G g(y-x) f(y) dm(y)?$$
$(3)$ Why $$ \int_G g(y-x)dm^{\prime}(x)= \int_G g(-x)dm^{\prime}(x) > 0?$$ I suppose the equality is due to translation invariant property of Haar measure? I have no idea on how to get $>0.$
For Question 1, apply Fubini's Theorem to the double integral $\iint g(y)f(x-y)\ dm'(x)\,dm(y)$. For Question 2, use the translation invariance of $m$, translating by $x$, to get $\int g(y)f(x+y))\,dm(y)=\int g(y-x)f(y)\,dm(y)$. For Question 3, I think $g$ should have been chosen to be non-negative at the beginning of the proof. Then it's actually positive somewhere (since its $m$-integral is positive) and therefore bounded below by some positive $\epsilon$ on some nonempty open set (because it's continuous), and therefore has positive $m'$-integral (because a Haar measure gives every nonempty open set positive measure).