Outer Measures on a Linear Lattice

In this paper we define and discuss the theory of abstract outer measures on a sequentially continuous(2) linear lattice S. This is a generalization of the concept of outer measure on a function space as used by Bourbaki [3]. H. Nakano [7] and M. H. Stone [8] have modernized Lebesgue's extension theory; our approach provides a common generalization of their theories. We show, for example, that their theories lead to the same set of integrable functions, and that, from a lattice theoretic point of view, Stone's results depend on the fact that the space of all real-valued functions on a set X is a perfect(3) lattice.