# Set-valued map (measurability)

I have this exercice and i want to know how to solve it :

1)- Let $X,Y$ two separable metric spaces ,let $(\Omega, \mathcal{A})$ be a measurable space ,and $f: \Omega \rightarrow X$ a measurable function and $\varphi :\Omega \times X \rightarrow Y$ a Cartheodory function ,then the map: $\Omega\rightarrow Y$, $x\mapsto \varphi (x,f(x))$ is measurable .

please; i don't know how to start .

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Look the book" fundamentals of applied functional analysis" authors : Dragisa Mitrovic and darko zubrinic . look at the page 57 . in this page have a theorem and this theorem is a particular case of your question . maybe the proof apresented there can help ( my english is terrible, sorry... ) –  math student Feb 25 '13 at 4:13
look the proof of the lemma 4.52 of this book : books.google.com.br/… maybe can help –  math student Feb 25 '13 at 4:21