Functions defined by sequences of integrals and the inversion of approximate derived numbers

Abstract

have been determined.t It is easy to construct sequences for which the limit on the left side of (1) exists and is different from the right side. In the present paper necessary and sufficient conditions are obtained for the existence of this limit in terms of the sequence s,(x). It turns out that the limit function F(x) is independent of f(x). In fact if F(x) is an arbitrary continuous function, there exists a sequence of summable functions sn(x) tending to zero everywhere, for which <sn(x)dx tends to F(x) everywhere. If Fl(x), F2(x), is a sequence of measurable functions, there is a sequence sn(x) tending to zero everywhere for which almost everywhere the set of limits of f sn(x)dx is the sequence Fl(x), F2(x), I f . . fIf esn(x)dx is bounded in n and e, e any measurable subset of (a, b), then F(x), when it exists, is of bounded variation on (a, b). Conversely, if F(x) is of bounded variation, there exists a function f(x) and a sequence of summable functions s (x) tending everywhere to f(x) for which f:Sn(x)dx tends to F(x), and for which fesndx is bounded in n and e. This is of some interest for the reason that it provides a characterization of functions of bounded variation which can be extended to functions of any number of variables. It is possible for the limit of the left side of (1) to exist when the function f(x) is not summable. As an aid in the study of this situation we introduce the following conventions: