We study non-linear functionals, including quasi-linear functionals, p-conic quasi-linear functionals, d-functionals, r-functionals, and their relationships to deficient topological measures and topological measures on locally compact spaces. We prove representation theorems and show, in particular, that there is an order-preserving, conic-linear bijection between the class of finite deficient topological measures and the class of bounded p-conic quasi-linear functionals. Our results imply known representation theorems for finite topological measures and deficient topological measures. When the space is compact we obtain four equivalent definitions of a quasi-linear functional and four equivalent definitions of functionals corresponding to deficient topological measures.

中文翻译：

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局部紧空间上的非线性泛函、不足的拓扑测度和表示定理

我们研究非线性泛函，包括拟线性泛函、p-圆锥拟线性泛函、d-泛函、r-泛函，以及它们与局部紧空间上的不足拓扑测度和拓扑测度的关系。我们证明了表示定理，并特别表明在有限缺陷拓扑测度类和有界 p 圆锥拟线性泛函类之间存在保序、圆锥线性双射。我们的结果意味着有限拓扑测度和缺陷拓扑测度的已知表示定理。当空间紧凑时，我们得到四个等价的拟线性泛函定义和四个对应于不足拓扑测度的泛函的等价定义。