an:05702413 Zbl 1198.46037 Halu\v{s}ka, J\'an; Hutn{\'\i}k, Ondrej The general Fubini theorem in complete bornological locally convex spaces. EN Banach J. Math. Anal. 4, No. 2, 53-74, electronic only (2010). ISSN 1735-8787/e 2010

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*46G10 28B05 Fubini theorem; bilinear integral; bornology; locally convex topological vector space; product measure; Dobrakov integral 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. Surjit Singh Khurana (Iowa City) Zbl 1184.46043