The aim of this paper is to show that if axiom M (or the continuum hypothesis) is assumed, then every upper semi-continuous compact-valued map from the space of irrationals to a compact (not necessarily metric) space has a selection, which is measurable in the sense that pre-images of Baire measurable sets are universally measurable. The methods used will yield generalizations and easier proofs of well-known theorems, namely of a selection theorem by Sion [1], and a representation theorem by Ioffe [3].
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