Let $ (X,\mathcal{F},\mu) $ and $ (G,\mathcal{G},\nu) $ be two measure spaces with $ \mu $ and $ \nu $ being $ \sigma $-finite. Per definition, the linear span of $$ \{ \mathbf{1}_{C} ~|~ C \in \mathcal{F} \otimes \mathcal{G} ~ \text{and} ~ (\mu \otimes \nu)(C) < \infty \} $$ is dense in $ {L^{p}}(X \times Y,\mathcal{F} \otimes \mathcal{G},\mu \otimes \nu) $ for any $ p \in [1,\infty) $. This should also be true for the linear span of $$ \{ \mathbf{1}_{A \times B} ~|~ A \in \mathcal{F}, ~ B \in \mathcal{G} ~ \text{and} ~ \mu(A),\nu(B) < \infty \}. $$ Do you know a reference?

Thank you.