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Transactions of the American Mathematical Society

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Weak-star convergence in the dual of the continuous functions on the $ n$-cube, $ 1\leq n\leq \infty $

Authors: Richard B. Darst and Zorabi Honargohar
Journal: Trans. Amer. Math. Soc. 275 (1983), 357-372
MSC: Primary 46E27; Secondary 26B30, 28A33, 60B10
MathSciNet review: 678356
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Abstract: Let $ n$ be a positive integer and let $ J = \times _{j = 1}^n{[0,1]_j}$ denote the $ n$-cube. Let $ \mathbf{C} = \mathbf{C}(J)$ denote the (sup norm) space of continuous (real-valued) functions defined on $ J$, and let $ \mathfrak{M}$ denote the (variation norm) space of (real-valued) signed Borel measures defined on the Borel subsets of $ J$. Let $ \left\langle {{\mu _l}} \right\rangle $ be a sequence of elements of $ \mathfrak{M}$. Necessary and sufficient conditions are given in order that $ {\text{li}}{{\text{m}}_l}\int f \,d{\mu _l}$ exists for every $ f \in \mathbf{C}$. After considering a finite dimensional case, the infinite dimensional case is entertained.

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Article copyright: © Copyright 1983 American Mathematical Society