If there is a composite function $\Phi: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ such that $\Phi(x,s) = u(x+hs)$, where $h$ is constant and $u$ is a measurable function from $\mathbb{R} \to \mathbb{R}$, then $\Phi$ is a measurable? (Assuming Lesbegue measures are given on $\mathbb{R}$ and $\mathbb{R}^2$)
The reason why I am asking this is that I am wondering if I can apply Fubini theorem to this $\Phi$. From Brezis' book Functional Analysis, it says the following.
$\int_\mathbb{R} dx \int^1_0 \vert u'(x+sh)\vert^p ds = \int^1_0 ds \int_\mathbb{R} \vert u'(x+sh)\vert^p dx$. (p.208)
I guess he applied the Fubini theorem here, but if we want to do that, it is necessary to make sure that $\vert u'(x+sh)\vert^p$ is measurable on $\mathbb{R} \times [0,1]$, where $u' \in L^p(\mathbb{R})$. I don't know how to prove it. Could you please help me? Thank you very much!
Let $g:\mathbb R\times\mathbb R\to\mathbb R\times\mathbb R$ be prescribed by $(x,s)\mapsto (x,hs)$.
Let $f:\mathbb R\times\mathbb R\to\mathbb R$ be prescribed by $(x,y)\mapsto x+y$.
Then both functions are continuous, hence Borel-measurable and we recognize that:$$\Phi=u\circ f\circ g$$As a composition of Borel-measurable functions also $\Phi$ is Borel-measurable.