SOME REMARKS ON INTEGRAL OPERATORS AND EQUIMEASURABLE SETS

We give a characterisation of equimeasurable sets in terms of the difference between the notions of almost everywhere convergence and convergence in measure. We apply this characterisation to obtain a direct proof of a criterion for integral representability of operators, due to A. V. Bukhvalov (obt...

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Bibliographic Details
Main Author: Schachermayer, Walter
Format: Book Chapter
Language:English
Subjects:
Bounded Subset
Compact Operator
Infinite Product
Integral Operator
Probability & statistics
Real analysis
Trace Class
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Summary:We give a characterisation of equimeasurable sets in terms of the difference between the notions of almost everywhere convergence and convergence in measure. We apply this characterisation to obtain a direct proof of a criterion for integral representability of operators, due to A. V. Bukhvalov (obtained in 1974) by a criterion of the present author (obtained in 1979). In the second part - following an idea due to A. Costé - we show that convolution with a suitably chosen singular measure defines a positive operator on L2, which is of trace class p, for p > 2, but fails to be integral. This sharpens a result, due to D. H. Fremlin.
ISSN:0075-8434
1617-9692
DOI:10.1007/BFb0076303