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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 $?

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.

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This was crossposted from math.SE: 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 math.stackexchange also… – user34108 May 18 '13 at 1:43

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