Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-15T18:21:32.051Z Has data issue: false hasContentIssue false
  • English
  • Français

Abstract integral spaces and minimal extensions

Published online by Cambridge University Press:  18 May 2009

M. J. Maron
M. J. Maron
Affiliation:
University of Glasgow
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The development of the theory of absolute integrals derives from certain key facts. Among them are:

(I) An integral is a positive linear functional on a vector lattice, which is continuous in a certain sense.

(II) A function equal almost everywhere to a summable function is itself summable.

(III) Every measurable function is the pointwise limit of a sequence of elementary step functions.

A device that often plays an important role in measure theory, but which has not beenfully exploited in the theory of abstract integrals is that of

(IV) the smallest class containing a given class and having a certain property

(such as being a σ-ring of sets). It is our purpose in this paper to examine the theory of abstract real-valued absolute integrals axiomatically, in such a way as to isolate and clarify the roles of (I) through (IV).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 12 , Issue 2 , September 1971 , pp. 166 - 178
Copyright
Copyright © Glasgow Mathematical Journal Trust 1971

References

REFERENCES

1.Bogdanowicz, W. M., An approach to the theory of integration generated by positive linear functionals and existence of minimal extensions, Proc. Japan Acad. 43 (1967), 186–191.Google Scholar
2.Daniell, P. J., A general form of integral, Ann. of Math. 19 (1918), 279–294.CrossRefGoogle Scholar
3.Loomis, L. H., An introduction to abstract harmonic analysis (New York, 1953).Google Scholar
4.Segal, I. E. and Kunze, R. A., Integrals and operators (New York, 1968).Google Scholar
5.Stone, M. H., Notes on integration I, II, III, IV, Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 336–342, 447–455, 483–490, 35 (1949), 50–58.CrossRefGoogle Scholar
6.Taylor, A. E., General theory of functions and integration (New York, 1965).Google Scholar
7.Terpe, F., Maximale eingelagerte absolutkonvergente Integrate, Math. Nachr. 28 (1965), 257–274.CrossRefGoogle Scholar
8.Terpe, F., Das Stone-integral als maximales eingelagertes absolutkonvergentes Integral, Math. Z. 107 (1968), 59–66.CrossRefGoogle Scholar
9.Young, W. H., A new method in the theory of integration, Proc. London Math. Soc. (2) 9 (1911), 15–50.CrossRefGoogle Scholar