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Vector lattices and regularizers

Published online by Cambridge University Press:  26 February 2010

Casper Goffman
Casper Goffman
Affiliation:
Purdue University and Westfield College, University of London.
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Extract

In this paper, we consider a class of spaces for which the convolutions with any set of regularizes converge in the topology of the space. We have already dealt with this matter in [2], but the conditions on the topology were unnecessarily restrictive and the proof somewhat unnatural. The present theorem is not only substantially more general, but is also more satisfying in that the argument reveals an unexpected connection between two topics; namely, the approximation of Lebesgue integrals by means of Riemann sums, and the uniqueness of certain types of locally convex topologies in vector lattices.

Type
Research Article
Information
Mathematika , Volume 15 , Issue 1 , June 1968 , pp. 63 - 69
Copyright
Copyright © University College London 1968

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References

1.Doob, J. L., Stochastic Processes, p. 63 (New York, 1953).Google Scholar
2.Goffman, C., “A property of integral means”, Duke Math. J., 24 (1957), 505–510.CrossRefGoogle Scholar
3.Goffman, C., “Compatible semi norms in a vector lattice”, Proc. Nat. Acad. Sci., 42 (1956), 536–538.CrossRefGoogle Scholar
4.Jessen, B., “On the approximation of Lebesgue integrals by Reimann sums”, Annals of Math., 35 (1934), 248–251.CrossRefGoogle Scholar
5.Kist, J., “Locally o-convex spaces”, Duke Math. J., 25 (1958), 569–582.CrossRefGoogle Scholar