Decomposable sets

Decomposability, a theory analogous to convexity, has recently achieved a major importance in optimal control and differential inclusions. A set M in the space of integrable functions L¹(T) is said to be decomposable if for all f, g ∈ M, χA·f + (1 −χA)·g ∈ M for every measurable subset A of T. Several results on convexity can be transferred to decomposability (e.g. existence of continuous selections, fixed points, and continuous perturbations), and they lead to nontrivial and useful results in optimization theory. Moreover, by integrating decomposable sets in L¹, we get an abstract formulation of the linear bang-bang principle, which is a certain generalization of Lyapunov's range theorem, too.