Extensions of abstract valued set functions
J. E. Huneycutt · Transactions of the American Mathematical Society · 1969
Chapter IIn measure theory, an essential concept is the extension of the notion of a "measure" on one class of sets to a notion of measure on a larger class of sets.This paper investigates such extensions when the "measure" takes values in a topological abelian group.We also investigate relationships between regularity and countable additivity.It is shown that under sufficiently strong regularity conditions, a modular function on a lattice determines a unique countably additive function on the semiring generated by the lattice.Similar theorems are derived showing that finitely additive functions which are sufficiently regular on a ring or semiring are also countably additive.The final chapter investigates the extension of a Banach space valued, modular function on a lattice to a countably additive function on the a-ring generated by the lattice.Chapter II Let X be a set and 3t a semiring of subsets of X; let S be an abelian group.A function fi : Jf -> 'S is additive provided that for each finite, pairwise disjoint sequence {Atfl in $f whose union is in ^ m(Ui ^i) = 2i M-^í)-Von Neumann [5, p. 94] showed that any additive function on a semiring has a unique additive extension to the smallest ring 0t(2tF) containing df.If 'S is a topological abelian group, we can define countable additivity of an additive function /x : 3tf ->■ IS by requiring that {2ï M^f)}w = i converges to /¿(U" -¿t) whenever {At}^ is a pairwise disjoint sequence of members of Mf whose union is a member of #P.Von Neumann also showed that if the function im: ¿F -> 'S is countably additive, then the unique additive extension to the ring 0t(2tf) is also countably additive.Thus it is of interest to know conditions under which an additive function on a semiring to a topological abelian group will be countably additive.One such condition ensuring countable additivity is regularity.By saying that ¡x : 3tif -> 'S is regular, we usually mean that each member of ^f can be "approximated from below by compact sets" and "approximated from above by open sets."Presented to the Society,