Extension of Multiplicative Set Functions with Values in a Banach Algebra

Let X be a given set, and let R, S be some classes of certain subsets of X. We say that R is a ring if A+B ∈ R, A−B ∈ R whenever A ∈ R, B ∈ R. We say that S is a σ-ring if it is a ring and if for every sequence A1, A2, . . ., where Ai ∈ S (i = 1, 2, . . .), the sum ∑∞ i=1 Ai is also contained in S. Let B be a commutative Banach algebra with a unity, i.e. a Banach space, where for all pairs f ∈ B, g ∈ B a product fg = gf is defined such that if h ∈ B, then (fg)h = f(gh), (f +g)h = fh+gh, ‖fg‖ ≤ ‖f‖ ‖g‖ and there is an e ∈ B such that ef = fe = f , ‖e‖ = 1. I shall consider set functions f(A) defined on the elements of a ring R such that the values f(A) lie in B; f(A) ∈ B for A ∈ R. A real-valued set function α(A) defined on R is called of bounded variation if there is a number K such that for every finite sequence of pairwise disjoint sets A1, A2, . . . , Ar, Ai ∈ R (i = 1, 2, . . . , r), we have