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Fubini theorems for Orlicz spaces of Lebesgue-Bochner measurable functions


Author: Vernon Zander
Journal: Proc. Amer. Math. Soc. 32 (1972), 102-110
MSC: Primary 46E30; Secondary 28A35
DOI: https://doi.org/10.1090/S0002-9939-1972-0291791-4
MathSciNet review: 0291791
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Abstract: Let $ (X,V,v)$ be the volume space formed as the product of the volume spaces $ ({X_i},{V_i},{v_i})\;(i = 1,2)$. Let $ (p,q)$ be a pair of complementary (continuous) Young's functions, let $ Y,Z,{Z_1},{Z_2}$, be Banach spaces, let w be a multilinear continuous operator on $ Y \times {Z_1} \times {Z_2} \to W$. Let $ {L_p}(v,Y)$ be the Orlicz space of Lebesgue-Bochner measurable functions generated by p, and let $ {K_q}(v,Z)$ be the associated space of finitely additive Z-valued set functions. The principal result of this paper is as follows: Let $ f \in {L_p}(v,Y), {\mu _2} \in {K_q}({v_2},{Z_2})$. Then (a) the function $ f({x_1}, \cdot )$ is $ {v_2}$-Orlicz summable $ {v_1}$-a.e.; (b) the operator $ r(f,{\mu _2})$ defined by the expression $ r(f,{\mu _2})({x_1}) = \smallint {w_1}(f({x_1},{x_2}),{\mu _2}(d{x_2}))\;{v_1}$-a.e. is bilinear and continuous from $ {L_p}(v,Y) \times {K_q}({v_2},{Z_2})$ into $ {L_p}({v_1},{Y_1})/N$, where $ {w_1}(y,{z_2}) = w(y,{z_2})$, where $ {Y_1}$ is the Banach space of bounded linear operators from $ {Z_1}$ into W, and where N is the set of $ {Y_1}$-valued $ {v_1}$-measurable functions of zero seminorm; (c) the equality $ \smallint w(f,d{\mu _1},d{\mu _2}) = \smallint {w_0}(r(f,{\mu _2}),d{\mu _1})$ holds for all $ f \in {L_p}(v,Y),{u_i} \in {K_q}({v_i},{Z_i})\;(i = 1,2)$, where $ {w_0}({y_1},{z_1}) = {y_1}({z_1})$ for all $ {y_1} \in {Y_1},{z_1} \in {Z_1}$.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0291791-4
Keywords: Fubini theorem, Orlicz spaces, finitely additive set functions, Lebesgue-Bochner measurable functions
Article copyright: © Copyright 1972 American Mathematical Society