## 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 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 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 linear operators 46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators 46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)

Zbl 0094.12003
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