Abstract
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.
Keywords
- Integral Operator
- Compact Operator
- Bounded Subset
- Trace Class
- Infinite Product
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© 1986 Springer-Verlag Berlin Heidelberg
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Schachermayer, W. (1986). Some remarks on integral operators and equimeasurable sets. In: Letta, G., Pratelli, M. (eds) Probability and Analysis. Lecture Notes in Mathematics, vol 1206. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0076303
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DOI: https://doi.org/10.1007/BFb0076303
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