Abstract
We prove a selection theorem for convex-valued lower semicontinuous mappings F to Fréchet spaces under the assumption that every value of F contains all interior (in the convex sense) points of its own closure. This is an extension of E. Michael's theorem 3.1‴ in [7] for mappings to Banach spaces. The desired continuous single-valued selection is constructed as a pointwise barycenter mapping with respect to a suitable family of probability measures concentrated on values of F. As an application, we show that, for any metric space M there is a continuous mapping which to every compact set K⊂M assigns a probability measure whose support coincides with K. Earlier this fact was proved for a complete separable metric space M by using a fundamentally different technique based on Milyutin mappings.
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