A Course on Borel Sets

Abstract

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Deduction Systems for Coalgebras Over Measurable Spaces

“…As topological spaces these are all Polish: separable and completely metrizable. But it is a theorem of Kuratowski that between any two uncountable Polish spaces there is a Borel isomorphism, a measurable bijection with measurable inverse (see [Sri98,Section 3.3]). Thus from the point of view of the category Meas, these classical spaces are all essentially the same object.…”

Causality for Nonlocal Phenomena

“…By the first property in (52), both integrals in (53) are nonnegative and hence they both must vanish. However, by the second property in (52), the integrand in the rightmost integral is positive on I + ; therefore, this integral cannot vanish unless ω(I + ) = 0.…”

Nucleation in the one-dimensional stochastic Cahn-Hilliard model

“…Recalling that in D(ρ) the energy E λ (u) essentially coincides with the quasi-potential I(h 0 , u), as shown in Lemma 4.1, we can then formulate the following main result of this paper. Theorem 4.2 Suppose that Assumption 2.7 holds, let λ ∈ Λ, let D(κ, ρ) be defined as in (40), and consider the exit time τ defined in (41). Then for any δ > κ we have…”