Vol. 35, No. 3, 1970

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Products of finitely additive set functions from Orlicz spaces

Vernon Emil Zander

Vol. 35 (1970), No. 3, 799–804

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
Received: 13 February 1970
Published: 1 December 1970
Vernon Emil Zander