existence and uniqueness for $\dot x(t)=f(t,x(t))$, with $f$ measurable in $t$. Consider the following scalar ODE
$$\dot x(t) = f(t,x(t))$$
defined on a compact interval $[0,T]$.
Assume that $f$ is Lipschitz continuous in $x$, but it is measurable in $t$. 
What can we say about the existence and uniqueness?
The standard Picard theorem assumes that $f$ is continuous in $t$.

The answer is almost provided by Carathéodory’s existence theorem.
