Product measures and $\sigma-$ finite measures

Question

Problem similar to folland chapter 2 problem 51.

The actual problem in Folland mentions that $X,Y$ are not necessarily $\sigma-$finite. Then how can I use Fubini-Tonelli theorem?

Answer 1

Every $L^1$ function vanishes outside of a $\sigma$-finite set $M_f$ (use $\{x\mid f(x)|\geq 1/n\}$).

Then apply Fubini on the set $M_f \times M_g$ only.