向量值积分的代数理论
Algebraic theory of vector-valued integration
摘要：
我们定义了一个单子M一类可测量的代数集，我们展示了如何产生这个单子的理论向量值的整合，是有关的概念佩蒂斯积分。我们证明了这个单子的代数X是一个代数学上的局部凸向量空间，它被赋予了将X中的向量\int f dm与传入映射f：T --> X和T上的测度m相关联的运算。证明了只要Banach空间中的每一个有界弱可测函数都有Pettis积分，则Banach空间是M-代数。因此，所有可分的巴拿赫空间和所有自反的巴拿赫空间都是M-代数。
英文摘要：
We define a monad M on a category of measurable bornological sets, and we show how this monad gives rise to a theory of vector-valued integration that is related to the notion of Pettis integral. We show that an algebra X of this monad is a bornological locally convex vector space endowed with operations which associate vectors \int f dm in X to incoming maps f:T --> X and measures m on T. We prove that a Banach space is an M-algebra as soon as it has a Pettis integral for each incoming bounded weakly-measurable function. It follows that all separable Banach spaces, and all reflexive Banach spaces, are M-algebras.
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