Ways of obtaining topological measures on locally compact spaces

2019

Anal Math

Self Cite

In this paper we define and study signed deficient topological measures and signed topological measures (which generalize signed measures) on locally compact spaces. We prove that a signed deficient topological measure is τ -smooth on open sets and τ -smooth on compact sets. We show that the family of signed measures that are differences of two Radon measures is properly contained in the family of signed topological measures, which in turn is properly contained in the family of signed deficient topological measures. Extending known results for compact spaces, we prove that a signed topological measure is the difference of its positive and negative variations if at least one variation is finite; we also show that the total variation is the sum of the positive and negative variations. If the space is locally compact, connected, locally connected, and has the Alexandroff one-point compactification of genus 0, a signed topological measure of finite norm can be represented as a difference of two topological measures.2010 Mathematics Subject Classification. Primary 28C15; Secondary 28C99. Key words and phrases. signed deficient topological measure; signed topological measure; positive, negative, total variation of a signed deficient topological measure.

Section: Proofmentioning

confidence: 81%

Section: Proofmentioning

confidence: 96%

“…] that contains f (X) and g(X)) we see that µ g (A) = µ h (A) for A = F or A = U . For ν g we apply a similar argument using formula (6).…”

Section: Proofmentioning

confidence: 99%

“…(As ν one may take, for instance, a topological measure from Example 1 or Example 2 in [4]). Then µ is a signed topological measure.…”

Section: Signed Topological Measures On a Locally Compact Spacementioning

confidence: 99%

Section: Signed Topological Measures On a Locally Compact Spacementioning

confidence: 99%

See 3 more Smart Citations

Signed Topological Measures on Locally Compact Spaces

2019

Anal Math

Self Cite

Weak Convergence of Topological Measures

2021

J Theor Probab

Topological measures and deficient topological measures are defined on open and closed subsets of a topological space, generalize regular Borel measures, and correspond to (nonlinear in general) functionals that are linear on singly generated subalgebras or singly generated cones of functions. They lack subadditivity, and many standard techniques of measure theory and functional analysis do not apply to them. Nevertheless, we show that many classical results of probability theory hold for topological and deficient topological measures. In particular, we prove a version of Aleksandrov’s theorem for equivalent definitions of weak convergence of deficient topological measures. We also prove a version of Prokhorov’s theorem which relates the existence of a weakly convergent subsequence in any sequence in a family of topological measures to the characteristics of being a uniformly bounded in variation and uniformly tight family. We define Prokhorov and Kantorovich–Rubenstein metrics and show that convergence in either of them implies weak convergence of (deficient) topological measures on metric spaces. We also generalize many known results about various dense and nowhere dense subsets of deficient topological measures. The present paper constitutes a necessary step to further research in probability theory and its applications in the context of (deficient) topological measures and corresponding nonlinear functionals.

Repeated quasi-integration on locally compact spaces

2022

Positivity

When X is locally compact, a quasi-integral (also called a quasi-linear functional) on $$ C_c(X)$$ C c ( X ) is a homogeneous, positive functional that is only assumed to be linear on singly-generated subalgebras. We study simple and almost simple quasi-integrals, i.e., quasi-integrals whose corresponding compact-finite topological measures assume exactly two values. We present equivalent conditions for a quasi-integral to be simple or almost simple. We give a criterion for repeated quasi-integration (i.e., iterated integration with respect to topological measures) to yield a quasi-linear functional. We find a criterion for a double quasi-integral to be simple or almost simple. We describe how a product of topological measures acts on open and compact sets. We show that different orders of integration in repeated quasi-integrals give the same quasi-integral if and only if the corresponding topological measures are both measures or one of the corresponding topological measures is a positive scalar multiple of a point mass.