Let $f$ be a non-negative measurable function in $\mathbb{R}$. Suppose that $\iint_{\mathbb{R}^2}{f(4x)f(x-3y)\,dxdy}=2\,.$ Calculate $\int_{-\infty}^{\infty}{f(x)\,dx}$.
My first thought was to change the order of integration so that I integrate in $y$ first, since there's only a single $y$ in the integrand... but I'm not sure how/if that even helps me.
Then the more I thought about it, the less clear it was to me that Fubini's theorem even applies as it's written. Somehow I need a function of two variables. So should I set $g(x,y) = f(4x)f(x-3y)$ and do something with that? At least Fubini's theorem applies for $g(x,y)$, since we know it's integrable on $\mathbb{R}^2$. .... Maybe?
I'm just pretty lost on this, so any help you could offer would be great. Thanks!