On some functionals

1. In this paper we intend to give a new proof and various generalizations of the following theorem due to Hahn :$ If If,(t) } is a sequence of summable functions in the interval J= (0, 1) and if lim. f Ef,(t)dt existsfor every measurable set E c J, then the indefinite integrals Fn(x) =fx fn(t)dt are equally absolutely continuous in J and therefore converge to an absolutely continuousfunction. The proof will be based on a theorem of Baire which has proved useful in many similar cases.? Incidentally there will be given a generalization of another theorem concerning sequences of functional transformations and published in a previous paper by the author.? 2. We shall denote by R the space of measurable characteristic functions in the interval J = (0, 1), i.e., functions which almost everywhere assume two values only, 0 and 1. The distance of two functions x(t), y(t) in R is defined by the formula