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 $\tau$-smooth on open sets and $\tau$-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.

中文翻译：

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局部紧空间的有符号拓扑测度

在本文中，我们定义并研究了局部紧空间上的有符号缺陷拓扑测度和有符号拓扑测度（推广了有符号测度）。我们证明了有符号缺陷拓扑测度在开集上是 $\tau$-smooth，在紧集上是 $\tau$-smooth。我们表明，作为两个氡测度差异的签名测度族正确包含在签名拓扑测度族中，而后者又正确包含在签名缺陷拓扑测度族中。扩展紧凑空间的已知结果，我们证明如果至少一个变化是有限的，则有符号拓扑测度是其正变化和负变化之差；我们还表明总变化是正变化和负变化的总和。如果空间是局部紧凑的，连通的，