Historical Survey

Abstract

This chapter contains a historical survey about limit and boundedness theorems for measures since the beginning of the last century. In these kinds of theorems, there are two substantially different methods of proofs: the sliding hump technique and the use of the Baire category theorem. We deal with Vitali-Hahn-Saks, Brooks-Jewett, Nikodým convergence and boundedness theorems, and we consider also some related topics, among which Hahn-Schur-type theorems and some other kind of matrix theorems, the uniform boundedness principle and some (weak) compactness properties of spaces of measures. In this context, the Rosenthal lemma, the biting lemma and the Antosik-Mikusiński-type diagonal lemmas play an important role. We consider the historical evolution of convergence and boundedness theorems for σ- additive, finitely additive and non-additive measures, not only real-valued and defined on σ-algebras, but also defined and/or with values in abstract structures.

Keywords: σ-additive measure, Baire category theorem, biting lemma, Brooks- Jewett theorem, D-poset, Drewnowski lemma, finitely additive measure, Hahn- Schur theorem, interpolation property, k-triangular set function, matrix theorem, MV-algebra, Nikodým boundedness theorem, Nikodým convergence theorem, orthomodular lattice, orthomodular poset, Rosenthal lemma, Sliding hump, Vitali set, Vitali-Hahn-Saks theorem.