Quasi-linear functionals on locally compact spaces

Abstract

This paper combines new and known results in a single convenient source for anyone interested in learning about quasi-linear functionals on locally compact spaces. We define singly generated subalgebras in different settings and study signed and positive quasi-linear functionals. Quasi-linear functionals are, in general, nonlinear, but linear on singly generated subalgebras. The paper gives representation theorems for quasi-linear functionals on C c (X), for bounded quasi-linear functionals on C 0 (X) on a locally compact space, and for quasi-linear functionals on C(X) on a compact space. There is an order-preserving bijection between quasi-linear functionals and compact-finite topological measures, which is also “isometric” when topological measures are finite. We present many properties of quasi-linear functionals and give an explicit example of a quasi-linear functional on ℝ 2 . Results of the paper will be helpful for further study and application of quasi-linear functionals in different areas of mathematics, including symplectic geometry.