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:


1. the function $f(x,t)$ be defined and continuous in $x$ for all most all t;
2. $f(x,t)$ be measurable in $t$ for each $x$;
3. $|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.



Caratheodory equations
You can find that in §10, XVIII of Walter,W.: Ordinary Differential Equations, volume 182 of Graduate Texts in Mathematics. Springer,
New York, 1998. Translated from the sixth German (1996) edition by Russell Thompson, Readings in Mathematics.


