Let $G$ be a locally compact group which is a countable union of compact subsets and $\lambda$ a left Haar measure on $G$ (i.e. $(G,\lambda)$ is $\sigma$-finite). Let $f,g \in L^1(G)$. Then I have a question on the following application of Tonelli/Fubini: $$\int_G \int_G |f(y)||g(y^{-1}x)|d\lambda(y)d\lambda(x)=\int_G \int_G |f(y)||g(y^{-1}x)|d\lambda(x)d\lambda(y)$$ Why is this legitimate? I do not quite see why $|f(y)||g(y^{-1}x)|$ is measurable which actually reduces to the question why the function of two variables $f(y)g(y^{-1}x)$ is measurable.

Let $G$ be a locally compact group which is a countable union of compact subsets and $\lambda$ a left Haar measure on $G$ (i.e. $(G,\lambda)$ is $\sigma$-finite). Let $f,g \in L^1(G)$. Then I have a question on the following application of Tonelli/Fubini: $$\int_G \int_G |f(y)||g(y^{-1}x)|d\lambda(y)d\lambda(x)=\int_G \int_G |f(y)||g(y^{-1}x)|d\lambda(x)d\lambda(y)$$ Why is this legitimate? I do not quite see why $|f(y)||g(y^{-1}x)|$ is measurable which actually reduces to the question why the function of two variables $f(y)g(y^{-1}x)$ is measurable.

Edit. This can be found in the third edition of the book classical fourier analysis by Loukas Grafakos, p. 20.