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The general Fubini theorem in complete bornological locally convex spaces. (English) Zbl 1198.46037

This paper is a continuation of the authors’ paper [“The Fubini theorem for bornological product measures”, Results Math. 54, No. 1–2, 65–73 (2009; Zbl 1184.46043)]. Here, X,Y,Z are Hausdorff complete bornological locally convex spaces with filtering upwards bases of bornologies 𝒰,𝒲,𝒱, respectively; here, each U𝒰 is a closed, absolutely convex bounded subset of X and UU 0 a fixed closed, absolutely convex bounded subset of X. The subspace of X generated by U, with Minkowski functional of U, is a Banach space X U . The topology of X is the inductive limit topology of the Banach spaces {X U :U𝒰}. Similar properties hold for the topologies of Y,Z arising from 𝒲,𝒱 respectively. L(X,Y) is the space of all continuous linear functions from X to Y; similarly for L(Y,Z) and L(X,Z).

T,S are two sets, and are δ-rings on T and S, respectively, and m:L(X,Y) and n:L(Y,Z) are two measures. With the help of (U,W,V)(𝒰,𝒲,𝒱), the authors reduce the study from complete bornological locally convex spaces X,Y,Z to the Banach spaces X U ,Y W ,Z V ,. Under certain conditions, an existence theorem for the product measure mn:L(X,Z) is proved. Then under certain conditions, a Fubini type theorem is proved for this product measure.

MSC:
46G10Vector-valued measures and integration
28B05Vector-valued set functions, measures and integrals (measure theory)