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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Continuous functions on multipolar sets


Author: Ramasamy Jesuraj
Journal: Proc. Amer. Math. Soc. 99 (1987), 331-339
MSC: Primary 31D05
MathSciNet review: 870796
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Abstract: Let $ \Omega = {\Omega _1} \times \cdots \times {\Omega _n}(n > 1)$ be a product of $ n$ Brelot harmonic spaces each of which has a bounded potential, and let $ K$ be a compact subset of $ \Omega $. Then, $ K$ is an $ n$-polar set with the property that every $ i$-section $ (1 \leqslant i < n)$ of $ K$ through any point in $ \Omega $ is $ (n - i)$ polar if and only if every positive continuous function on $ K$ can be extended to a continuous potential on $ \Omega $. Further, it has been shown that if $ f$ is a nonnegative continuus function on $ \Omega $ with compact support, then $ MRf$, the multireduced function of $ f$ over $ \Omega $, is also a continuous function on $ \Omega $.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1987-0870796-7
PII: S 0002-9939(1987)0870796-7
Article copyright: © Copyright 1987 American Mathematical Society