Let $\Omega$ be a smooth bounded domain. If $u \in L^2(\Omega \times \{0,1\})$, then $$\int_{\Omega \times \{0,1\}}|u(x,y)|^2 < \infty$$.
How to interepret the integral $\int_{\Omega \times \{0,1\}}f(x,y)$? I expect it to be equal to $\int_{\Omega} f(x,0) + \int_{\Omega} f(x,1)$, but I don't know if it is true.
Edit: I think we can write $\Omega \times \{0,1\} = \Omega \times \{0\} \cup \Omega \times \{1\}$ and then split the integral which gives us $$\int_{\Omega \times \{0\}}f(x,y) + \int_{\Omega \times \{1\}}f(x,y)$$ and then..?