I would like to verify my solution for the following problem:

Let $g:(0,\mathrm{\infty})\to \mathbb{R}$ be a nonzero, continuous function and define $G:[0,1]\times [1,\mathrm{\infty})\to \mathbb{R}$ by

$G(x,y)=g(xy).$

Show that $G$ is not in ${L}^{1}([0,1]\times [1,\mathrm{\infty}))$.

Solution:

${\int}_{1}^{\mathrm{\infty}}{\int}_{0}^{1}|G(x,y)|dxdy={\int}_{1}^{\mathrm{\infty}}{\int}_{0}^{1}|g(xy)|dxdy={\int}_{1}^{\mathrm{\infty}}{\int}_{0}^{y}\frac{{\textstyle |g(t)|}}{{\textstyle y}}dtdy\ge {\int}_{1}^{\mathrm{\infty}}\frac{{\textstyle 1}}{{\textstyle y}}\underset{t\in (0,y]}{inf}|g(t)|\phantom{\rule{thickmathspace}{0ex}}dy\ge \underset{t\in (0,\mathrm{\infty})}{inf}|g(t)|{\int}_{1}^{\mathrm{\infty}}\frac{{\textstyle 1}}{{\textstyle y}}dy=\mathrm{\infty}$

I'm a little concerned about taking the infimum over the non compact set (0,y] (g is not defined at 0).

Is this correct? And if so, how do I address my concern above? Thanks for your time.