Let $M = (\mathbb{R}, \mathcal{L}, \ell)$ be the D5268: Real lebesgue measure space such that

(i)	$M_{\times} = (\mathbb{R} \times \mathbb{R}, \mathcal{L} \times \mathcal{L}, \ell \times \ell)$ is a D2706: Product measure space of $M$ with itself
(ii)	$f : \mathbb{R} \times \mathbb{R} \to \mathbb{C}$ is an D1921: Absolutely integrable function on $M_{\times}$

Let $g, h : \mathbb{R} \to \mathbb{C}$ each be a D4881: Complex function such that

(i)	\begin{equation} \forall \, x, y \in \mathbb{R} : f(x, y) = g(x) h(y) \end{equation}

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} $\square$