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.

  • $\begingroup$ The title translates as "Differential equations in the real field, part 2". I did not find an online copy of it, and I suspect it would take quite a while to wait for the library to get it-otherwise I would transcribe (and translate) here the exact statement and its proof directly from Sansone's. $\endgroup$ Aug 19, 2014 at 8:52

2 Answers 2


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.


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).


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