Approximation of vector valued functions
It is known that a lower semi-continuous function from a metric space to \([0,\infty)\) can be approximated by an increasing sequence of Lipschitz continuous functions. This is generalized to a class of vector lattice valued functions. For a complete Banach lattice \(E\), an extension \(E^*\) is considered by adjoining an ``\(\infty\)''. An appropriate type of continuity, called inf-continuity, is considered in proving approximation results. The Lipschitz approximation leads to the consideration of a variational inferior limit and a convergence concept called \(V\)-convergence. Various results describing convergence are provided and these results are applied to obtain stochastic theorems.
scientific article; zbMATH DE number 1121766