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.



Simple functions on a product measure space
A useful density lemma is the following. 



> 
> Let $(X,\mathcal{A}, \mu)$ be a
>  measure space and let $\Gamma\subset\mathcal{P}(X)$ a ring of sets
>  of finite measure that generates the
>  $\sigma$-algebra $\mathcal{A}$. Then,
>  the linear span of the characteristic
>  functions of sets in $\Gamma$ is dense
>  in $L^p$ (here $1\le p < +\infty$)
> 
> 
> 


In your case, of course, you can take $\Gamma$ to be the collection of finite unions of Cartesian products of measurable sets of finite measure.


Incidentally, the hypothesis on $\Gamma$ in that density lemma can be weakened (one does not need all the ring strucure of $\Gamma$). Let's say that $\Gamma\subset \mathcal{P}(X)$ is a "semi-ring" (*warning*: not standard; I borrowed it from Halmos, with a slightly more general meaning) if the following holds: 



> 
> For all $A$ and $B$ in $\Gamma$ the
>  sets $A\setminus B$ and $A\cap B$ are
>  both expressible as union of 
>  countably many disjoint element of
>  $\Gamma$.
> 
> 
> 


Then the above lemma holds true. The notion of "semi-ring"is also interesting, in that it is a convenient domain for a completely additive set function, in order that the Caratheodory's Extension Theorem holds. 


