局部紧空间上关于亏拓扑测度的积分
Integration with respect to deficient topological measures on locally compact spaces
摘要：
拓扑测度和亏拓扑测度是Borel测度的推广，对应于某些非线性泛函.研究了局部紧空间上关于亏拓扑测度的积分。如果我们积分一个在无穷远处消失的非负函数，则这种集合上的积分产生一个新的亏拓扑测度;如果我们在紧空间上积分一个连续函数，则它产生一个带符号的亏拓扑测度。我们提出了许多性质，这些由此产生的亏损拓扑措施和签署亏损拓扑措施。特别地，它们关于原亏拓扑测度是绝对连续的，并且是Lipschitz连续的。由集合积分得到的亏拓扑测度也可以由非线性泛函得到。我们证明了，对于一个亏拓扑测度$ μ$，假设许多值，有一个功能$ f $，$\int_X f \, d μ= 0$，但$\int_X (-f )\, d μ\neq 0$。我们提出了不同的标准为$\int_X f \, d μ= 0$。我们还证明了一些收敛性结果，包括一个单调收敛定理。
英文摘要：
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 vanishing at infinity function; and it produces a signed deficient topological measure if we use 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 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 $\int_X f \, d μ= 0$, but $\int_X (-f )\, d μ\neq 0$. We present different criteria for $\int_X f \, d μ= 0$. We also prove some convergence results, including a Monotone convergence theorem.
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