Abstract
An integral is defined for not necessarily bounded functions, with respect to a finitely additive, possibly infinite measure. In particular this strictly includes the familiar case of absolutely convergent improper Riemann integration in Euclidean space. However, the measure need not have a countably additive extension, and no topology is assumed on the underlying space. The “Jordan field,” of sets with integrable characteristic functions, is used to characterize the integrable functions as those that are “nearly” Jordan measurable, and their integrals are shown to equal the (Lebesgue) integrals of the measures of the appropriate Jordan sets.
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