Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 1198.46037
Haluška, Ján; Hutn{\'\i}k, Ondrej
The general Fubini theorem in complete bornological locally convex spaces. (English)
[J] Banach J. Math. Anal. 4, No. 2, 53-74, electronic only (2010). ISSN 1735-8787/e

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)]

MSC 2000:: *46G10 Vector-valued measures and integration
28B05 Vector-valued set functions etc. (measure theory)

Keywords: Fubini theorem; bilinear integral; bornology; locally convex topological vector space; product measure; Dobrakov integral

Citations: Zbl 1184.46043

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]