Korovkin-type theorems for a countably sublinear functional (Q1116044)

The Korovkin closure Kor(H) of a subset H of a Banach lattice E is the set of all \(f\in H\) satisfying lim \(T_{\alpha}f=f\) for every equicontinuous net \(\{T_{\alpha}\}\) of positive linear operators on E such that lim \(T_{\alpha}h=h\) for all \(h\in H.\) Let X be a locally compact Hausdorff space with a countable base and J the set of all Borel measurable extended real-valued functions on X. Using the notion of a countable sublinear functional \(\gamma\) on J it is possible to define a space \({\mathcal L}(\gamma)\) which for specific \(\gamma\) reduces to \(C_ 0(X)\) or \({\mathcal L}_ p(X).\) Korovkin-type theorems then characterize the Korovkin closure of subsets of \({\mathcal L}(\gamma)\). These theorems generalize known results for \(C_ 0(X)\) and \({\mathcal L}_ p(X)\).