dense subspace of $L^2(\Omega\times(0,T))$

Question

I am trying to prove that the functions $f(\omega,t)=g(\omega)h(t)$ where $g\in G,\: h\in H,$ are dense in $L^2(\Omega\times(0,T))$ if $G$ is dense in $L^2(\Omega)$ and $H$ is dense in $L^2((0,T))$. If I fix one of the variables then i find a sequence of approximating functions in the other variable, thus for all t I have a sequence $g_n(\omega)\to f(\omega,t)$ in $L^2(\Omega)$ but how can I make it work even with the other variable? I would appreciate it if anyone could help me. Thank you in advance.

Answer 1

Hints: Here, I'll assume that the measure $\mu$ on $(\Omega,\mathcal{A})$ is finite.

1. Since elementary functions are dense in $L^2(\Omega \times (0,T))$ it suffices to show the claim for any elementary function, i.e. a function of the form $$f(\omega,t) = \sum_{j=1}^n c_j 1_{C_j}(\omega,t)$$ where $C_j \in \mathcal{A} \otimes \mathcal{B}([0,T])$ are elements of the product $\sigma$-algebra on $\Omega \times (0,T)$. Using linearity, we can restrict ourselves to indicator functions: $$f(\omega,t) = 1_{C}(\omega,t)$$ for some $C \in \mathcal{A} \otimes \mathcal{B}([0,T])$.
2. Show that $$\mathcal{D} := \{C \in \mathcal{A} \otimes \mathcal{B}((0,T)): \text{the claim holds for} \, f=1_C\}$$ defines a Dynkin system which is stable under intersections.
3. By assumption, $\mathcal{A} \times \mathcal{B}((0,T)) \subseteq \mathcal{D}$. Consequently, $$\mathcal{A} \otimes \mathcal{B}((0,T)) = \sigma(\mathcal{A} \times \mathcal{B}((0,T))) \subseteq \sigma(\mathcal{D}) = \mathcal{D}.$$