Applying results

(i)	R4373: Fubini's theorem for a basic real lebesgue measure space
(ii)	R2425: Linearity of complex Lebesgue integral

we have \begin{equation} \begin{split} \int_{\mathbb{R} \times \mathbb{R}} f \, d (\ell \times \ell) & = \int_{\mathbb{R}} \left( \int_{\mathbb{R}} f(x, y) \, \ell(d y) \right) \, \ell(d x) \\ & = \int_{\mathbb{R}} g(x) \left( \int_{\mathbb{R}} h(y) \, \ell(d y) \right) \, \ell(d x) \\ & = \left( \int_{\mathbb{R}} g(x) \, \ell(d x) \right) \left( \int_{\mathbb{R}} h(y) \, \ell(d y) \right) \\ \end{split} \end{equation} $\square$