Let $\Omega\subset \mathbb{R}^n$ be bounded and let $X:=H^1(\Omega)$. Let $a:\Omega\times \mathbb{R} \to \mathbb{R}, (x,z)\mapsto a(x,z)$ be a bounded function such that $a(x,.)$ is continuous on $\mathbb{R}$ for every $x$ and $a(.,z)$ is measurable on $\Omega$ for every $z$.
For $v,\phi\in H^1$ one finds in pde the integral $$\int_\Omega a(x,v(x))\nabla v(x) \cdot \nabla\phi(x) dx$$ All measures should be the Lebesgue measure My question is: Why is $a(x,v(x))$ a measurable function, why is $a(x,z)$ measurable on $\Omega\times \mathbb{R}$ or why is the integral defined?
Usually one can not conclude from measurability of each compontent to measurability on the product space. Somehow i feel I need more measure theory than i know yet :)