Density of $\mathcal{C}_c(A\times B)$ in $L^p(A, L^q(B))$ Let $A, B$ be two open sets in  $\mathbb{R}^n, \mathbb{R}^m$ respectively and denote $\mathcal{C}_c(A\times B)$ the  space of continuous functions with compact support in $A\times B.$ 
Is $\mathcal{C}_c(A\times B)$ dense in $L^p(A, L^q(B))$ for any $+\infty > q,p \geq 1 ?$ 
I believe that the answer is YES and I'm looking for a simple proof. It is well known that  $\mathcal{C}_c(A\times B)$ is dense in $L^r(A\times B)$ 
for any $r\geq 1.$ Does this help?
Thank you for any hint.
 A: Here is a more general statement. Suppose $V\subset L^p(A)$ and $W\subset L^q(B)$ are dense subspaces. Define $V\otimes W$ to be their algebraic tensor product, that is the linear span of functions $F(x,y)=f(x)g(y)$ for $f\in V$, $g\in W$. 
Theorem: $V\otimes W$ is dense in $L^p(A, L^q(B))$  provided $p,q\in [1,\infty)$. 
To apply this theorem to your case, let $V$ and $W$ be spaces of continuous functions with compact support; then $V\otimes W\subset C_c(A\times B)$.
Proof of the theorem: Recall that a function $\phi\in L^p(A,L^q(B))$ is simple if there is a partition of $A$ into $A_1,\dots,A_N$ such that 
$$
\phi = \sum_{k=1}^N \chi_{A_k} g_k\quad \text{for some } g_k\in L^q(B) \tag1
$$
Consider $f\in L^p(A,L^q(B))$. Since $f$ is Bochner measurable, there exists a sequence $(f_n)$ of simple functions that converges to $f$ pointwise. Using the density of $W$, we may arrange so that the constituent functions $g_k$ in (1) are from $W$. 
Consider the sets
$$E_n = \{x \in A : \|f_n(x)\|_{L^q} \le 2\|f(x)\|_{L^q}\}$$
Sine $\|f_n(\cdot)\|_{L^q} \to \|f(\cdot)\|_{L^q}$ pointwise, each point of $x$ where $f(x)\ne 0$ is eventually in $E_n$. Hence,  letting 
$$
h_n = f_n \chi_{E_n}
$$
we still have $h_n\to f$ pointwise. But now 
$$
\|h_n(x)-f(x)\|_{L^q}\le \|h_n(x)\|_{L^q}+ \|f(x)\|_{L^q} \le 3\|f(x)\|_{L^q}
$$
so by the dominated convergence theorem $h_n\to f$ in $L^p(A,L^q(B))$. 
So far we have shown that functions of the form 
$$
\phi = \sum_{k=1}^N \chi_{A_k} g_k\quad \text{for some } g_k\in W \tag2
$$
are dense in $L^p(A,L^q(B))$. But each such $\phi$ can be approximated by elements of $V\otimes W$ by picking a sequence from $V$ that converges to $\chi_{A_k}$ in $L^p(A)$. $\quad\Box$
The above is essentially Proposition 1.2 on page 2 of the book Martingales in Banach spaces by Pisier, an early draft of which can be found on author's page.
