*(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, $\u25b3$ and $\u25bd$ are $\delta $-rings on $T$ and $S$, respectively, and $m:\u25b3\to L(X,Y)$ and $n:\u25bd\to L(Y,Z)$ 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 $m\otimes n:\u25b3\otimes \u25bd\to L(X,Z)$ 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) |