Integral of multivalued mappings and its connection with differential relations. (English) Zbl 0536.28006

Jarník, Jiří; Kurzweil, Jaroslav

Čas. Pěst. Mat. 108, 8-28 (1983).

The authors of this paper present a Riemann type definition for integrals of multivalued functions and study its properties. The definition given by them is substantially different from that given by Z. Artstein and J. A. Burns [Pac. J. Math. 58, 297-307 (1975; Zbl 0324.28006)]. Let \(F(t)=[0,1]\) for \(t\in [a,b]\) (a compact interval), then \((B)\quad \int_{T}F(t)dt\) exists, while the (B)-integral of \(G(t)=F(t)\cup \{- \chi_ M(t)\},\) where \(\chi_ M\) is the characteristic function of a non-measurable set \(M\subset T\) does not exist in the sense of Artstein and Burns. This seemed rather unusual and hence the necessity of a change in the definition was felt by the authors. A function (multivalued) \(F:T\to {\mathcal S}^ n\) (\({\mathcal S}^ n\) denotes the family of all subsets of \({\mathbb{R}}^ n\) and T a compact interval) is integrable bounded if there is an integrable function \(\rho:T\to [0,+\infty)\) such that \(F(t)\subset \bar B(0,\rho(t))\) (closed ball with center 0 and radius \(\rho\) (t)) for a.e. \(t\in T\). The authors have studied mostly integrably bounded functions in this paper and kept more complete results for the future.

Reviewer: M.K.Nayak

MSC:

28B20	Set-valued set functions and measures; integration of set-valued functions; measurable selections
26A42	Integrals of Riemann, Stieltjes and Lebesgue type

Keywords:

differential relations; integrals of multivalued functions

Citations:

Zbl 0324.28006