given $f\in L^+(X,\mathcal{F})$ and $g\in L^+(X,\mathcal{G})$ then $h(x,y)=f(x)\cdot g(y)\in (X\times Y, \mathcal{F}\otimes\mathcal{G})$

Question

given $f\in L^+(X,\mathcal{F})$ and $g\in L^+(X,\mathcal{G})$ then $h(x,y)=f(x)\cdot g(y)\in (X\times Y, \mathcal{F}\otimes\mathcal{G})$

I was trying to show that $h^{-1}((a,\infty))=\{(x,y):f(x)\cdot g(y)\in (a,\infty)\}\in \mathcal{F}\otimes\mathcal{G}$ using the fact $\mathcal{B_\mathbb{R}}= \sigma(\{(a,\infty):a\in \mathbb{R}\})$ But couldn't make it.