A Theory of Integration

Extract

The usual development of the Lebesgue integral starts with a measure that may have been derived from some simpler set function, its associated class of measurable sets, the corresponding set of measurable functions, and operations which ultimately define the integral of any given function of this class, except for certain ones which are unbounded above and below. Here we propose to define a process of integration with respect to a set function more general than a measure. This process allows us to integrate virtually all functions real-valued on our space. The integrals thus obtained are all completely additive on a certain completely additive class of sets. Under rather mild hypotheses, we are able to delineate this class of sets explicitly.