Q:

a question about caratheodory condition

$f:[a,b]\times\mathbb{R}\rightarrow\mathbb{R}$ 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?

A:

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

