Remark 2

Applied to Lebesgue measure and to regulated functions, Prop. 3 of No. 3 yields anew the theorem on passage to the limit for integrals of regulated functions on a compact interval (FRV, II, §3, No. 1, Prop. 1); for sequences (or for filters with a countable base) of regulated functions, Th. 2 of No. 3 greatly improves on this proposition since, for uniformly bounded regulated functions on a compact interval, it substitutes pointwise convergence for uniform convergence (cf. §5, No. 4, Th. 2). However, as regards the passage to the limit for absolutely convergent integrals of regulated functions on a noncompact interval, we note that the conditions of Th. 2 of No. 3 imply that the integrals under consideration are uniformly convergent (in the sense defined in FRV, II, §3, No. 2), and thus do not improve on the conditions for convergence given in Book IV (loc. cit.) except as concerns the convergence of the functions $f_\alpha$ on every compact interval. Finally, the conditions for passing to the limit given for non absolutely convergent integrals of regulated functions remain outside the theory developed in this chapter.