Weak and point-wise convergence in for filter convergence |
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Authors: | Vladimir Kadets Alexander Leonov |
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Institution: | aDepartment of Mechanics and Mathematics, Kharkov National University, pl. Svobody 4, 61077 Kharkov, Ukraine |
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Abstract: | We study those filters on for which weak -convergence of bounded sequences in C(K) is equivalent to point-wise -convergence. We show that it is sufficient to require this property only for C0,1] and that the filter-analogue of the Rainwater extremal test theorem arises from it. There are ultrafilters which do not have this property and under the continuum hypothesis there are ultrafilters which have it. This implies that the validity of the Lebesgue dominated convergence theorem for -convergence is more restrictive than the property which we study. |
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Keywords: | Measurable functions Filter convergence Dominated convergence theorem for filters Extremal test for weak convergence Banach space |
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