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This note establishes two
results on products of finitely additive vector-valued set functions from Orlicz spaces.
A triple (Ω,Σ,μ) 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 (Ωi,Σi,μi)(i = 1,⋯,n) a family of charge spaces and (Ω,Σ,μ) 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
AΦ(Ωi,Σi,μi,Zi) belongs to the Orlicz space AΦ(Ω,Σ,μ,W).
For the infinite product case the following result holds:
Theorem. For (Ωt,Σt,μt)(t ∈ T) a family of probability charge spaces and
(Ω,Σ,μ) the product probability charge space, for u an infinitely linear bounded
operator on the multiplicative product space PT(AΦ(Ωt,Σt,μt,Zt),vt′) the
function v0 defined by v0(A) = u(v(A)) for A ∈ Σ belongs to the Orlicz space
AΦ(Ω,Σ,μ,W).
These results allow one to develop an integral determined by a product of charges
from Orlicz spaces.
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