On volumes generating the same lebesgue-bochner integration.

Introduction.-Let R,Y be the space of reals and a Banach space, respectively. The norm of elements in these spaces will be denoted by |. A nonempty family of sets V of an abstract space X will be called a pre-ring if for any two sets A,,A2a V we have Ai nA2G V, and there exist disjoint sets B1,..., BkEV such that A1\A2 = B1U ... UBt. A nonnegative finite-valued function v on the pre-ring V will be called a volume if for every countable family of disjoint sets A EV(tC T) such that A = U A EV we have v(A) = 2v(A,). T T In an earlier paper,' a direct construction of the space L(v,Y) of LebesgueBochner summable functions was presented and the theory of an integral of the form fu(fd/u) was developed. In the case when the bilinear operator is given by u(y,z) = zy for yC Y, zER, and JA = v, the above integral coincides with the classical Lebesgue-Bochner integral ffdv. All basic theorems concerning the algebraical and topological structures of the space L(v,Y) have been proved without developing the theory of measure or the theory of measurable functions. Basing the theory of integration on set functions defined on pre-rings, it was possible2y 3 to develop the theory of multilinear vectorial integration and to find integral representations of multilinear continuous operators on products of the spaces of Lebesgue-Bochner summable functions. It also permitted us to give new constructions of Fubini's theorem and to find its further generalizations.4 The theory of Lebesgue-Bochner measurable functions corresponding to the approach developed in reference 1 has been presented in reference 5. The theory of measure has been obtained as a by-product of the theory of integration. These results permitted us to simplify the theory of integration on locally compact spaces.6 It was also shown in reference 7 that integration generated by positive functionals can be easily obtained from integration generated by volumes. In a recent paper" the operation of completion of a volume v to a volume v, was introduced. In the present paper, using the above operation, we shall characterize volumes generating the same Lebesgue-Bochner integration. It will be also shown that any volume v on a pre-ring V has a unique extension to a measure on any a-ring W such that VcWcM, whereM denotes the family of all v-measurable sets. These results will be used to prove that integration generated by the product volume coincides with the integration generated by the product measure.'0 In reference 9 it will be shown that similar relations hold for vectorial integration. This in turn will imply that Fubini's theorems obtained in reference 4 represent generalization of the classical Fubini theorems. §1. A Necessary and Sufficient Condition for the Space of Summable Functions L(v, Y) to Be a Subspace of L(w, Y).-If v,w are two volumes defined respectively on