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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
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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

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