Vol. 35, No. 3, 1970

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ISSN: 0030-8730
Products of finitely additive set functions from Orlicz spaces

Vernon Emil Zander

Vol. 35 (1970), No. 3, 799–804
Abstract

This note establishes two results on products of finitely additive vector-valued set functions from Orlicz spaces. A triple ,Σ) is called a charge space if Σ is a ring of subsets of a set Ω and the charge μ is a finitely additive, nonnegative, finite-valued function with domain Σ.

Theorem. For i,Σii)(i = 1,,n) a family of charge spaces and ,Σ) the corresponding product charge space, for u an n-linear continuous operator from the product of the Banach spaces Z1,,Zn into a Banach space W, the function v defined by v(A) = u(vI(A),,vn(A)) for A Σ and vi from the Orlicz space AΦi,Σii,Zi) belongs to the Orlicz space AΦ,Σ,μ,W).

For the infinite product case the following result holds:

Theorem. For t,Σtt)(t T) a family of probability charge spaces and ,Σ) the product probability charge space, for u an infinitely linear bounded operator on the multiplicative product space PT(AΦt,Σtt,Zt),vt) the function v0 defined by v0(A) = u(v(A)) for A Σ belongs to the Orlicz space AΦ,Σ,μ,W).

These results allow one to develop an integral determined by a product of charges from Orlicz spaces.

Mathematical Subject Classification
Primary: 46.35
Milestones
Received: 13 February 1970
Published: 1 December 1970
Authors
Vernon Emil Zander