Simple functions on a product measure space 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.
 A: 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. 
A: All indicator functions of the algebra generated by rectangles are linear combinations of indicator functions of rectangles. Indicator functions of product measurable sets can be approximated by indicator functions of the algebra, which follows from the standard Caratheodory construction. Clearly, the same can be done with linear combinations of indicator functions.
