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 sets 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 sets. These positive, negative, and total variations turn out to be deficient topological measures. Then we examine finite additivity, superadditivity, smoothness, and other properties of deficient topological measures. We obtain methods for generating new deficient topological measures. We provide necessary and sufficient conditions for a deficient topological measure to be a topological measure and to be a measure. The results presented are necessary for further study of topological measures, deficient topological measures, and corresponding non-linear functionals on locally compact spaces.

中文翻译：

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局部紧空间上的拓扑测度不足

拓扑测度和拟线性泛函概括了测度和线性泛函。我们定义并研究了局部紧凑空间上的缺陷拓扑测度。局部紧空间上的一个缺陷拓扑测度是开集和闭集上的集合函数，它在紧集上是有限可加的，在开集上是内正则，在闭集上是外正则。不足的拓扑测度泛化了测度和拓扑测度。首先，我们研究了一个有符号集函数的正、负和总变化，该函数仅被假定为在紧集上是有限可加的。这些正的、负的和总的变化被证明是有缺陷的拓扑测量。然后我们检查有限可加性、超可加性、平滑性和其他缺陷拓扑测度的性质。我们获得了生成新的有缺陷的拓扑度量的方法。我们为有缺陷的拓扑测度成为拓扑测度和成为测度提供了充要条件。所呈现的结果对于进一步研究局部紧空间上的拓扑测度、缺陷拓扑测度和相应的非线性泛函是必要的。