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