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Another ‘short’ proof of the Riesz representation theorem

Published online by Cambridge University Press:  24 October 2008

D. J. H. Garling
D. J. H. Garling
Affiliation:
St John's College, Cambridge
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Extract

1. Introduction. In [2], a short synthetic proof of the Riesz representation theorem was given; this used the Hahn-Banach theorem, the Stone-Čech compactification of a discrete space and the Caratheodory extension procedure for measures. In this note, we show how the theorem can be proved using ultrapowers in place of the Stone-Čech compactification. We also describe how the proof can be expressed in a non-standard way (a rather different non-standard proof has been given by Loeb [4]).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 99 , Issue 2 , March 1986 , pp. 261 - 262
Copyright
Copyright © Cambridge Philosophical Society 1986

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References

REFERENCES

[1]Cutland, N. J.. Nonstandard measure theory and its applications. Bull. London Math. Soc. 15 (1983), 529–589.CrossRefGoogle Scholar
[2]Garling, D. J. H.. A ‘short’ proof of the Riesz representation theorem. Proc. Cambridge Philos. Soc. 73 (1973), 459–460.CrossRefGoogle Scholar
[3]Henson, C. W.. Analytic sets, Baire sets and the standard part map. Canad. J. Math. 31 (1979), 663–672.CrossRefGoogle Scholar
[4]Loeb, P.. A functional approach to non-standard measure theory. Contemporary Mathematics 26 (1984), 251–261.CrossRefGoogle Scholar