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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 $?

Obs: Let $H: \Omega \rightrightarrows \mathbb {R}^{n} $ and $(\Omega,\mathcal{F})$ be a measurable space. $H$ is $\mathcal{F}$-measurable if for each compact set $K \subset \mathbb {R}^{n}$, $$ \{x \in \Omega: H(x) \cap K \not= \emptyset\} \in \mathcal{F}. $$

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What notion of measurability for multifunctions do you use? –  Michael Greinecker May 17 '13 at 22:27
    
Castaing-Valadier –  user78308 May 17 '13 at 22:30
    
This was crossposted to MO. In the future, please wait some time before posting your question in multiple fora, and when you do, provide links to the other posts - as you can imagine, it would be frustrating for someone to put time into answering your question here, only to see hear from you that you'd already gotten the solution elsewhere. –  Zev Chonoles May 18 '13 at 1:18
    
I asked this question in the mathoverflow also mathoverflow.net/questions/131012/measurable-multifunction –  user78308 May 18 '13 at 1:39

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