The Integrability of a Sequence of Functions
R. L. Jeffery · Transactions of the American Mathematical Society · 1931
Introduction.Let f=fx,f2, • ■ • be a sequence of functions summable on the measurable set E, and convergent on E to the summable function F.the sequence is said to be integrable.If the above equality holds when the set E is replaced by any measurable part of E the sequence is said to be completely integrable.These questions of integrability and complete integrability have received considerable attention from various writers, f It has been shown by VitaliJ that the equi-convergence § of the sequence of integrals is both necessary and sufficient for complete integrability.It would then follow that this condition is sufficient for integrability.One can, however, easily construct examples which show that it is not necessary.The chief aim of the present paper is to determine conditions which are both necessary and sufficient for the integrability of the sequence/.This is accomplished by methods which yield, as special cases, some of the results already obtained by Vitali and de la Vallée Poussin. Definitions and preliminary results. In what follows, without again making mention of it, we shall use e to denote any measurable sub-set of E. We further define S {l, r¡), r/>0 and arbitrary, to be the part of E for which \F-fn | <v {n^l), and C{1, r¡) its complement on E; e{l, n)+ and e{l, «)_ the parts of C{1, y) for which /"^0,/"<0 respectively.It is easily verified that these sets are measurable, and that mC{l, v) tends to zero as I becomes infinite.Finally we use g = gx, g2, ■ • ■ to denote a sub-sequence/""/»" • • • of/.We shall have occasion to use