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Finitely-additive, countably-additive and internal probability measures

Haosui Duanmu, William Weiss (2018)

Commentationes Mathematicae Universitatis Carolinae

We discuss two ways to construct standard probability measures, called push-down measures, from internal probability measures. We show that the Wasserstein distance between an internal probability measure and its push-down measure is infinitesimal. As an application to standard probability theory, we show that every finitely-additive Borel probability measure P on a separable metric space is a limit of a sequence of countably-additive Borel probability measures { P n } n ∈ ℕ in the sense that ∫ f d P = lim n → ∞ ∫ f d P n for all bounded...

Gap universality of generalized Wigner and β -ensembles

László Erdős, Horng-Tzer Yau (2015)

Journal of the European Mathematical Society

We consider generalized Wigner ensembles and general β -ensembles with analytic potentials for any β ≥ 1 . The recent universality results in particular assert that the local averages of consecutive eigenvalue gaps in the bulk of the spectrum are universal in the sense that they coincide with those of the corresponding Gaussian β -ensembles. In this article, we show that local averaging is not necessary for this result, i.e. we prove that the single gap distributions in the bulk are universal. In fact,...

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