Caratheodory equations

Question

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.

Answer 1

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.

Answer 2

The Carathéodory hypotheses have been though exactly to this end. Note that:

• If $x_k(t)$ converges a.e. to $x(t)$, then $f(x_k(t), t)$ converges a.e. to $f(x ( t), t)$, by (1).

• If $u$ is a simple function, say $u(t) =\sum_{i=1}^m c_i \chi_{A_i}(t)$ for some finite measurable partition $\{A_i\}_i$ of $[a,b]$, then $f(u(t),t)=\sum_{i=1}^m f(c_i, t)\chi_{A_i}(t)$ is measurable, by (2).

• If $x(t)$ is any measurable function on $[a,b]$ then it is an a.e. limit of simple functions, so by (1) and (2) the function $f(x ( t),t)$ is measurable, in fact summable, by (3).