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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: http://books.google.com/books?id=4hIq6ExH7NoC&pg=PA153&lpg=PA153&dq=caratheodory+function&source=bl&ots=p9oRVwiZPD&sig=7cT9I9F2afZBX4wC7SxeiqQy8OI&hl=en&sa=X&ei=9hqJT6W1GZChtweA1PXZCQ&ved=0CC0Q6AEwAg#v=onepage&q=caratheodory%20function&f=false suffice?2012-04-14

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