Linear Functionals and Integrals in Abstract Spaces

In his addendum to Saks, Theory of the Integral, Banach considers a Lebesgue integral defined in a manner quite similar to that of Daniell and remarks that no use is made of a measure. I t is, however, quite easy to show that Banach's and Daniell's integrals are expressible as Lebesgue integrals whose measure functions are regular outer measures in the sense of Carathéodory. In the first two sections a linear, non-negative functional is considered. Upper and lower functionals are associated with this functional, and by means of them inner and outer measures are defined. I t is shown that if the inner and outer measures of a set coincide, the set is measurable. To establish the converse a continuity assumption is made in §3, and a representation theorem in terms of the Lebesgue integral is obtained. I t is shown in §4 that the theorem of Lebesgue for term-wise integration holds for semi-uniformly convergent 2Rsystems.