Ok, I am reading Fillipov book on discontinuous right hand side differential equations (the red book).
He states the next lemma: " Let the function $f(t,x)$ satisfy the Caratheodory conditions and let the function $x(t)$ for $a\leq t \leq b$ be measurable. Then the composite function $f(x(t),t)$ is summable."
Where Caratheordory codntions are: In the domain $D$ of the $(t,x)-space$, let:
- the function $f(x,t)$ be defined and continuous in $x$ for all most all t;
- $f(x,t)$ be measurable in $t$ for each $x$;
- $|f(x,t)|\leq m(t) $, the function $m(t)$ being summable (on each finite interval if $t$ is not bounded in $D$).
Now the book provides the next reference which seems to be not in English. Sansone , G., Equazioni Differenziali nel campo Reale. Parte 2.
Even though I can guess what the title says (differential equations... real ... part 2 ) :-)
Anyone has an English reference of the very exact theorem?
Thanks in advance.