# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 $U\in 𝒰$ 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 $\left\{{X}_{U}:U\in 𝒰\right\}$. Similar properties hold for the topologies of $Y,Z$ arising from $𝒲,𝒱$ respectively. $L\left(X,Y\right)$ is the space of all continuous linear functions from $X$ to $Y$; similarly for $L\left(Y,Z\right)$ and $L\left(X,Z\right)$.

$T,S$ are two sets, $△$ and $▽$ are $\delta$-rings on $T$ and $S$, respectively, and $m:△\to L\left(X,Y\right)$ and $n:▽\to L\left(Y,Z\right)$ are two measures. With the help of $\left(U,W,V\right)\in \left(𝒰,𝒲,𝒱\right)$, 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 $m\otimes n:△\otimes ▽\to L\left(X,Z\right)$ is proved. Then under certain conditions, a Fubini type theorem is proved for this product measure.

##### MSC:
 46G10 Vector-valued measures and integration 28B05 Vector-valued set functions, measures and integrals (measure theory)