For two continuous functions $f(x)$ and $g(y)$ let their integrals be given by:
$$I_1 = \int f(x)\,\mathrm{d}x$$
$$I_2 = \int g(y)\,\mathrm{d}y$$
Then under which conditions is it true that
$$\int \int f(x)\cdot g(y)\,\mathrm{d}y\,\mathrm{d}x = I_1 \cdot I_2$$
That is to say that the integral of their product is equal to the product of their integrals?
Whenever $\displaystyle\iint|f(x)g(y)|<\infty$. By Tonelli, you can check if both $\displaystyle\int|f(x)|<\infty$ and $\displaystyle\int|g(y)|<\infty$.
The condition is true when,
$$I_1=\int f(x) \mathrm{d}x$$ and $$I_2=\int g(y) \mathrm{d}y$$ exist and both are finite then,
$$I_1\cdot I_2= \int\int g(y)f(x)\mathrm{d}x\mathrm{d}y$$
Must exist.
