Signed topological measures on locally compact spaces

Decompositions of signed deficient topological measures

This paper focuses on various decompositions of topological measures, deficient topological measures, signed topological measures, and signed deficient topological measures. These set functions generalize measures and correspond to certain non-linear functionals. They may assume $\infty$ or $-\infty$. We introduce the concept of a proper signed deficient topological measure and show that a signed deficient topological measure can be represented as a sum of a signed Radon measure and a proper signed deficient topological measure. We also generalize practically all known results that involve pro…

关于无限能量符号Radon测度伪扫除的Fuglede问题

On Fuglede's problem on pseudo-balayage for signed Radon measures of infinite energy

针对局部紧空间上的适当核，发展一般符号Radon测度（含无限能量）到一般集合的内（外）伪扫除理论，解决Fuglede提出的理论扩展问题，并给出在加权最小能量问题中的应用。

Deficient topological measures on locally compact spaces

Topological measures and quasi-linear functionals generalize measures and linear functionals. We define and study deficient topological measures on locally compact spaces. A deficient topological measure on a locally compact space is a set function on open and closed subsets which is finitely additive on compact sets, inner regular on open sets, and outer regular on closed sets. Deficient topological measures generalize measures and topological measures. First we investigate positive, negative, and total variation of a signed set function that is only assumed to be finitely additive on compact…

Topological stability of continuous functions with respect to averaging by measures with locally constant densities

Let $μ$ be a measure on $[-1,1]$. Then for every continuous function $f:\mathbb{R}\to\mathbb{R}$ and $α>0$ one can define its averaging $f_α:\mathbb{R}\to\mathbb{R}$ by the formula: \[ f_α(x) = \int_{-1}^{1} f(x+tα)dμ. \] In arXiv:1509.06064 the authors studied the problem when $f_α$ is topologically equivalent to $f$ for all $α>0$ and call this property a topological stability of $f$ under averagings with respect to measure $μ$. Similarly one can define topological stability of a germ of $f$ at some point $x\in\mathbb{R}$. It was shown that for a continuous function $f:\mathbb{R}\to\mathbb{R}…

Characterizations of signed measures in the dual of $BV$ and related isometric isomorphisms

We characterize all (signed) measures in $BV_{\frac{n}{n-1}}(\mathbb{R}^n)^*$, where $BV_{\frac{n}{n-1}}(\mathbb{R}^n)$ is defined as the space of all functions $u$ in $L^{\frac{n}{n-1}}(\mathbb{R}^n)$ such that $Du$ is a finite vector-valued measure. We also show that $BV_{\frac{n}{n-1}}(\mathbb{R}^n)^*$ and $BV(\mathbb{R}^n)^*$ are isometrically isomorphic, where $BV(\mathbb{R}^n)$ is defined as the space of all functions $u$ in $L^{1}(\mathbb{R}^n)$ such that $Du$ is a finite vector-valued measure. As a consequence of our characterizations, an old issue raised in Meyers-Ziemer [MZ] is resol…