Integration with respect to deficient topological measures on locally compact spaces

Abstract

Topological measures and deficient topological measures generalize Borel measures and correspond to certain non-linear functionals. We study integration with respect to deficient topological measures on locally compact spaces. Such an integration over sets yields a new deficient topological measure if we integrate a nonnegative continuous vanishing at infinity function; and it produces a signed deficient topological measure if we integrate a continuous function on a compact space. We present many properties of these resulting deficient topological measures and of signed deficient topological measures. In particular, they are absolutely continuous with respect to the original deficient topological measure, and their corresponding non-linear functionals are Lipschitz continuous. Deficient topological measures obtained by integration over sets can also be obtained from non-linear functionals. We show that for a deficient topological measure μ that assumes finitely many values, there is a function f such that $\begin{array}{} \int\limits_X \end{array}$f dμ = 0, but $\begin{array}{} \int\limits_X \end{array}$ (–f) dμ ≠ 0. We present different criteria for $\begin{array}{} \int\limits_X \end{array}$f dμ = 0. We also prove some convergence results, including a Monotone convergence theorem.