On Additive and Multiplicative Totals

In the present paper terminology “total” is used for a generalization of the Burkill integral and multiplicative integral, respectively. The functions, the totals of which are considered, take their values from a Banach algebra B with a unity. This means a Banach space B in which for every pair f ∈ B, g ∈ B a product fg ∈ B is defined such that ‖fg‖ ≤ ‖f‖ ‖g‖ and if h ∈ B, then f(g + h) = fg + fh, (g + h)f = gf + hf , finally there is an e ∈ B with the properties ef = fe = f , ‖e‖ = 1. It is proved that under some conditions the additive (and multiplicative) total of a multiplicative (and additive, resp.) set function exists. The theorems of this type are useful in solving some functional equations (see e.g. [3] and § 6) and studying the properties of multiplicative set functions by tracing the problems to those formulated in terms of additive set functions.