Approximative compactness of linear combinations of characteristic functions

Best approximation by the set of all n-fold linear combinations of a family of characteristic functions of measurable subsets is investigated. Such combinations generalize Heaviside-type neural networks. Existence of best approximation is studied in terms of approximative compactness, which requires convergence of distance-minimizing sequences. We show that for (Omega, mu) a measure space, in L-p(Omega, mu) with 1 <= p <= infinity and for all n >= 1, compact families of characteristic functions of sets (of finite measure for p < infinity) generate approximatively compact n-fold linear spans. Results are illustrated by examples of continuously parametrized sets. (C) 2020 Elsevier Inc. All rights reserved.