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$f:[a,b]\times\mathbb{R}\rightarrow\mathbb{R}$ is an caratheodory function if

$(a)$ the map $z\rightarrow f(t,z)$ is continuous for almost all $t\in[a,b],$

$(b)$ the map $t\rightarrow f(t,z)$ is measurable for all $z\in\mathbb{R}$,

then $(a)(b)$ implies for $t\in[a,b]$ that $g(t,u(t))$ is measurable for any measurable u(t).

How to prove this result?

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Does Lemma 4.52 here:… suffice? – tchakravarty Apr 14 '12 at 6:38
thank you for your help – Jim Apr 14 '12 at 8:08

Hint: consider a sequence of simple functions $s_n$ converging to $u$ a.e. and use $(a), (b)$

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