Riesz整表示理论
Riesz integral representation theory
摘要：
我们提出了一个Riesz整表示理论，其中的功能，运营商和措施采取的值在一致交换幺半群（一个交换幺半群的一致性，使二元运算的幺半群一致连续）。它描述了该理论可以应用的算子和它们生成的有限可加性测度。满足这些条件的算子称为“Riesz积分”。给定一个基本的"Riesz系统“，证明了每一个Riesz积分都生成一种可加测度，这里称之为”Riesz测度“. Riesz积分和Riesz测度之间的对应是一个双射。一个简单的计算表明，如果一个算子具有这样的表示，那么它必须具有Hammerstein性质。对于拓扑向量空间，该理论给出了具有Hammerstein性质的算子是Riesz积分的充要条件。我们注意到，当考虑集值函数时，一致交换幺半群自然出现，并且Riesz系统的公理排除了某些无限可微函数的空间。
英文摘要：
We present a Riesz integral representation theory in which functions, operators and measures take values in uniform commutative monoids (a commutative monoid with a uniformity making the binary operation of the monoid uniformly continuous). It describes the operators to which the theory can be applied and the finitely-additive measures they generate. Operators satisfying the conditions will be called ``Riesz integrals''. Given an underlying ``Riesz system'', it is shown that every Riesz integral generates a certain kind of finitely additive measure called here a ``Riesz measure''. The correspondence between Riesz integrals and Riesz measures is a bijection. A straightforward calculation shows that if an operator has such a representation, then it must have the Hammerstein property. For topological vector spaces, the theory yields necessary and sufficient conditions for operators with the Hammerstein property to be Riesz integrals. We note that uniform commutative monoids arise naturally when considering set-valued functions, and that the axioms of a Riesz system rule out certain spaces of infinitely differentiable functions.
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