Published online by Cambridge University Press: 18 May 2009
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).