Convergence in Variation and Related Topics

Anthony P. Morse
1937 Transactions of the American Mathematical Society  
1. Introduction. In recent papersj by Adams and Clarkson and by Adams and Lewy the notions of convergence in variation and convergence in length have been examined. In AC it has been shown that if a sequence converges in variation and satisfies certain further restrictions which are clearly needed, the sequence of reciprocals converges in variation. The central purpose of the present paper is to determine so far as we are able the transformations which when applied to sequences of functions,
more » ... serve various types of convergence, such as convergence in variation or length and other types which we shall introduce. This paper also leads us to certain generalizations of results in AC and AL, such as Theorem 5.4 wherein convergence in length is seen to be invariant under addition and multiplication when only one of the limit functions is absolutely continuous. In §2 we assemble certain preliminary definitions, notations, and conventions. §3 is devoted to preliminary theorems and lemmas, among them being Theorems 3.1 and 3.2 which might be of interest in themselves, their full power, in fact, not being used in this paper. Theorem 3.2 is a substitution theorem for Lebesgue integrals which is more general than other theorems of this type known to us in literature. Certain results in §3 are, however, obvious analogues of results in AC. Transforms of sequences are discussed in §4, wherein Theorems 4.1 and 4.2 form the kernel of the paper. The remainder of the paper consists largely of various applications of these two theorems, convergence in length being discussed in §5 together with convergence almost in the mean, uniform convergence in length in §6, and strong convergence in §7. Certain miscellaneous applications are made in §8; these include Theorem 8.1 which points out a necessary and sufficient condition for convergence in the mean, and Theorems 8.3 and 8.4 which are generalizations of a theorem
doi:10.2307/1989878 fatcat:zhuwccoilrfwlcc7zkve76lbj4