Multifunctions on abstract measurable spaces and application to stochastic decision theory

Summary The main results are some very general theorems about measurable multifunctions on abstract measurable spaces with compact values in a separable metric space. It is shown that measurability is equivalent to the existence of a pointwise dense countable family of measurable selectors, and that the intersection of two compact-valued measurable multifunctions is measurable. These results are used to obtain a Filippov type implicit function theorem, and a general theorem concerning the measurability of y(t)=min f({t} × Γ(t)) when f is a real valued function and Γ a compact valued multifunction. An application to stochastic decision theory is given generalizing a result of Benes. The research in this paper was partially supported by University of Kansas General Research Fund Grants 3918-5038 and 3199-5038. Entrata in Redazione il 20 dicembre 1972.