We construct a homeomorphism between the compact regular locale of integrals on a Riesz space and the locale of measures(valuations) on its spectrum. In fact, we construct two geometric theories and show that they are biinterpretable. The constructions are elementary and tightly connected to the Riesz space structure. 2000 Mathematics Subject Classification 06D22, 28C05
Vickers @cite_19 presents another variant of the Riesz representation theorem. His construction works for locales which are not necessarily compact completely regular. However, his integrals have their values in the lower (or upper) reals, as opposed to the Dedekind reals. A locale of valuations was first presented by Heckman @cite_7 .
@cite_19 @cite_7
2023829897 2006686635 2144063313 2550342065
We introduce a new method to compute explicit formulae for various zeta functions associated to groups and rings. The specific form of these formulae enables us to deduce local functional equations. More precisely, we prove local functional equations for the subring zeta functions associated to rings, the subgroup, conjugacy and representation zeta functions of finitely generated, torsion-free nilpotent (or T -)groups, and the normal zeta functions of T -groups of class 2. In particular we solve the two problems posed in [9, Section 5]. We deduce our theorems from a ‘blueprint result’ on certain p-adic integrals which generalises work of Denef and others on Igusa’s local zeta function. The Malcev correspondence and a Kirillov-type theory developed by Howe are used to ‘linearise’ the problems of counting subgroups and representations in T -groups, respectively. Symmetric spaces over local fields, and the harmonic analysis of their class one representations, arise as the local calculation of Jacquet's theory of the relative trace formula. There is an extensive literature at the real place, but few general results for p-adic fields are known. The objective here is to carry over to symmetric spaces as much as possible of Queens and the prerequisite results of Howe, and to provide counterexamples for those things which do not generalize. We derive the Weyl integration formula, local constancy of the spherical character on the θ-regular set, Howe's conjecture for θ-groups, the germ expansion for spherical characters at the origin, and a spherical version of Howe's Kirillov theory for compact p-adic groups. We find that the density property of regular orbital integrals fails. Some of the basic ideas are nascent in Hakim's thesis (written under Herve Jacquet). An account of lower and upper integration is given. It is constructive in the sense of geometric logic. If the integrand takes its values in the nonnegative lower reals, then its lower integral with respect to a valuation is a lower real. If the integrand takes its values in the non-negative upper reals, then its upper integral with respect to a covaluation and with domain of integration bounded by a compact subspace is an upper real. Spaces of valuations and of covaluations are deflned. Riemann and Choquet integrals can be calculated in terms of these lower and upper integrals. Author(s): Johnson, William Andrew | Advisor(s): Scanlon, Thomas | Abstract: This dissertation is a collection of results in model theory, related in one way or another to fields, NIP theories, and elimination of imaginaries. The most important result is a classification of dp-minimal fields, presented in Chapter 9. We construct in a canonical fashion a non-trivial Hausdorff definable field topology on any unstable dp-minimal field. Usingthis we classify the dp-minimal pure fields and valued fields up to elementary equivalence. Furthermore we prove that every VC-minimal field is real closed or algebraically closed.In Chapter 11, we analyze the theories of existentially closed fields with several valuations and orderings, as studied by van den Dries. We show that these model complete theories are NTP2, and analyze forking, dividing, and burden in these theories. The theory of algebraically closed fields with n independent valuation rings turns out to be an example of such a theory. This provides a new and natural example of an NTP2 theory which is neithersimple nor NIP, nor even a conceptual hybrid of something simple and something NIP.In Chapter 8, we exhibit a bad failure of elimination of imaginaries in a dense o-minimal structure. We produce an exotic interpretable set which cannot be put in definable bijection with a definable set, after naming any amount of parameters. However, we show that these exotic interpretable sets are still amenable to some of the tools of tame topology: they mustadmit nice definable topologies locally homeomorphic to definable sets.Chapter 12 proves the existence of Z nZ-valued definable strong Euler characteristics on pseudofinite fields, which measure the non-standard “size” of definable sets, mod n. The non-trivial result is that these “sizes” are definable in families of definable sets. This could probably be proven e cohomology, but we give a more elementary proof relying heavily on the theory of abelian varieties.We also present simplified and new proofs of several model-theoretic facts, including the definability of irreducibility and Zariski closure in ACF (Chapter 10), and elimination of imaginaries in ACVF (Chapter 6). This latter fact was originally proven by Haskell, Hrushovski, and Macpherson. We give a proof that is drastically simpler, inspired by Poizat’s proofs of elimination of imaginaries in ACF and DCF.