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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 $\mathcal{U}, \mathcal{W}, \mathcal{V}$, respectively; here, each $ U \in \mathcal{U}$ is a closed, absolutely convex bounded subset of $X$ and $ U \supset U_{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 $\{ X_{U}: U \in \mathcal{U} \}$. Similar properties hold for the topologies of $Y, Z$ arising from $\mathcal{W}, \mathcal{V}$ 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, $\bigtriangleup$ and $\bigtriangledown$ are $\delta$-rings on $T$ and $S$, respectively, and $m: \bigtriangleup \to L(X, Y)$ and $n: \bigtriangledown \to L(Y, Z)$ are two measures. With the help of $(U, W, V) \in (\mathcal{U}, \mathcal{W}, \mathcal{V})$, 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 $m \otimes n : \bigtriangleup \otimes \bigtriangledown \to L(X, Z)$ is proved. Then under certain conditions, a Fubini type theorem is proved for this product measure.

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