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Extension theorems of continuous random linear operators on random domains. (English) Zbl 0879.47018
The author formulates and proves the complete version of random generalizations of the Hahn-Banach extension theorem for a continuous random linear operator defined on $Gr E$, where $Gr E$ is the graph of a measurable multifunction $E: \Omega\to B$, $B$ being a separable Banach space. For the case where $B$ is not separable, an analogous result is also formulated and proved. For proving these extension theorems, the author develops in the present paper some new techniques on so-called random normed module and related module homomorphisms. By this way the results of the author cover and refine results of {\it Y. L. Hou} (1991) as well as are reduced in the case $E(s) =M$ (linear subspace of $B)$ to the well-known results of {\it O. Hanš} [Ann. Math. Statistics 30, 1152-1157 (1959; Zbl 0094.12003)] $(B$ being separable) and L. H. Zhu (1988) $(B$ being not-separable).

##### MSC:
 47B80 Random operators (linear) 46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators 46H25 Normed modules and Banach modules, topological modules
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