The sustained failure of efforts to design an infinite lottery machine using ordinary probabilistic randomizers is traced back to a problem familiar to set theorists: we have no constructive prescriptions for probabilistically non-measurable sets. Yet construction of such sets is required if we are to be able to read the result of an infinite lottery machine that is built from ordinary probabilistic randomizers. All such designs face a dilemma: they can provide an accessible (readable) result with probability zero; or an inaccessible result with probability greater than zero.
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