Question about this integral $\int \left| g\left( y\right) \right| \int \left| f\left( x-y\right) \right| dx\,dy$

Question

I have seen this in my script without proof:

$$\int \left| g\left( y\right) \right| \int \left| f\left( x-y\right) \right| dx\,dy = \int \left| g\left( y\right) \right| dy\int \left| f\left( x\right) \right| dx.$$

Is this true?

Answer 1

Consider change of variable $x' = x-y$. Then for any $y$, $$\int |f(x-y)|\, dx = \int |f(x')| \, d x' = \int |f(x)|\, dx.$$ Therefore, $$\int |g(y)| \left(\int |f(x-y)|\, dx\right)dy = \int |g(y)| \left(\int |f(x)|\, dx\right)dy.$$ Observe that $\int |f(x)|\, dx$ is a constant, so we can pull it out of the integral and obtain the desired result.