Decomposable sets in the Lebesgue-Bochner spaces. (English) Zbl 0679.46032

Papageorgiou, Nikolaos S.

Comment. Math. Univ. St. Pauli 37, No. 1, 49-62 (1988).

Summary: Let X be a Banach space and \(K\subseteq L^ 1(X)\). We say that K is “decomposable” if for all measurable sets A and all \(f_ 1,f_ 2\in K\), \(\chi_ Af_ 1+\chi_{A^ c}f_ 2\in K\). In this paper we study the properties of such sets. So we obtain a new result about weak compactness in \(L^ 1(X)\), we study the image of decomposable sets under the action of continuous, linear operators and we derive some useful new properties of the set valued integral and of the set of integrable selectors of a measurable multifunction. Those results are then used to study convex integral functionals and to obtain a new “bang-bang” type theorem for infinite dimensional, linear control systems.

MSC:

46G10	Vector-valued measures and integration
46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
28B20	Set-valued set functions and measures; integration of set-valued functions; measurable selections
93C05	Linear systems in control theory
46A50	Compactness in topological linear spaces; angelic spaces, etc.

Keywords:

weak compactness; decomposable sets; set valued integral; set of integrable selectors of a measurable multifunction; convex integral functionals; bang-bang; infinite dimensional, linear control systems