Carathéodory conditions

If one wants to relax the continuity assumption on a function while preserving the natural equivalence between the Cauchy problem for the differential equation and the integral equation which can be obtained by integrating the Cauchy problem, one can follow ideas of C. Carathéodory [a1] and make the following definition.

Let be an open set and , . One says that satisfies the Carathéodory conditions on , written as , if

1) is measurable for every (cf. also Measurable function);

2) is continuous for almost every ;

3) for each compact set the function

is Lebesgue integrable (cf. also Lebesgue integral) on , where is the norm in .

If is a non-compact interval, one says that satisfies the local Carathéodory conditions on if for every compact interval . This is written as .

Note that any function which is the composition of and a measurable function , i.e. (cf. also Composite function), is measurable on .

To specify the space of the majorant more precisely, one says that is -Carathéodory, , if satisfies 1)–3) above with .

One can see that any function continuous on is -Carathéodory for any .

Similarly, one says that is locally -Carathéodory on if restricted to is -Carathéodory for every compact interval .

References

[a1]	C. Carathéodory, "Vorlesungen über reelle Funktionen" , Dover, reprint (1948)
[a2]	E. Coddington, N. Levinson, "The theory of ordinary differential equations" , McGraw-Hill (1955)
[a3]	M.A. Krasnoselskij, "Topological methods in the theory of nonlinear integral equations" , Pergamon (1964)
[a4]	J. Kurzweil, "Ordinary differential equations" , Elsevier (1986)
[a5]	A.F. Filippov, "Differential equations with discontinuous right hand sides" , Kluwer Acad. Publ. (1988)