Vector lattices and additive set functions
Kôsaku Yosida · Proceedings of the Japan Academy Series A Mathematical Sciences · 1941
1. In$'oduction.One of the fundamental theorems in the theory of the integral is that of Radon-Nikodym concerning the countably ad- ditive set functions.The proof given below ( 2) of this theorem will be shorter than the standard one in S. Saks' book or the one recently published by C. Carath6odory".Our proof is carried out by a maximal method, by making use of a lemma which is a simple modification of H. Hahn's decomposition theorem".The same method is also applied ( 3 and 4) to give lattice-theoretic formulations of Radon-Nikodym's theorem.Of these ten years, such formulations were given more or less explicitly by many authors,