Let $f:[a,b]\times \mathbb{R}^{n} \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}$. Suppose $ f (.,x, u) $ is Lebesgue measurable for each $(x,u)$. Suppose also that $ f $ is continuous at $ (x, u) $ for each fixed $t$.
Let $ g: [a, b] \rightarrow [a, b] $ be a Lebesgue measurable function and $ U: [a, b] \rightrightarrows \mathbb {R}^{m} $ be a Lebesgue measurable multifunction. Then the multifunction $$ F(t,x) = f(g(t),x,U(t)) $$ é $\mathcal{L}\times \mathcal{B}^{n}$-measurable (Lebesgue-Borel)? Or should require continuity in $ t $ in the function $ f $?
Note: Let $ T: \Omega \rightrightarrows \mathbb{R}^{n}$ and $(\Omega,\mathcal{F})$ a space measurable. $T$ is $\mathcal{F}$-measurable if for each compact set $K \subset \mathbb{R}^{n}$, $$ {x \in \Omega: T(x)\cap K \not= \emptyset} $$ is a $\mathcal{F}$-measurable set.
