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 ∫X $\begin{array}{} \int\limits_X \end{array}$ f dμ = 0, but ∫X $\begin{array}{} \int\limits_X \end{array}$ (–f) dμ ≠ 0. We present different criteria for ∫X $\begin{array}{} \int\limits_X \end{array}$ f dμ = 0. We also prove some convergence results, including a Monotone convergence theorem.

中文翻译：

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局部紧空间上拓扑测度不足的集成

摘要 拓扑测度和缺陷拓扑测度概括了Borel测度并对应于某些非线性泛函。我们研究了局部紧凑空间上拓扑测量不足的集成。如果我们对无穷函数处的非负连续消失进行积分，则这种对集合的积分会产生一个新的有缺陷的拓扑测度；如果我们在紧致空间上对一个连续函数进行积分，它会产生一个带符号的缺陷拓扑测度。我们展示了这些由此产生的缺陷拓扑测度和有符号缺陷拓扑测度的许多性质。特别地，它们相对于原始的缺陷拓扑测度是绝对连续的，并且它们对应的非线性泛函是 Lipschitz 连续的。也可以从非线性泛函中获得通过集合积分获得的缺陷拓扑度量。我们证明，对于假设有限多个值的有缺陷的拓扑测度 μ，有一个函数 f 使得 ∫X $\begin{array}{} \int\limits_X \end{array}$ f dμ = 0，但是 ∫X $\begin{array}{} \int\limits_X \end{array}$ (–f) dμ ≠ 0。我们为 ∫X $\begin{array}{} \int\limits_X \end{array} 提出了不同的标准$ f dμ = 0。我们也证明了一些收敛结果，包括单调收敛定理。