Abstract
We introduce the notion of analytically and subanalytically tame mea- sures. These are measures which behave well in the globally subanalytic context; they preserve tameness: integrals of globally subanalytic functions with param- eters resp. analytic functions with parameters restricted to globally subanalytic compact sets are definable in an o-minimal structure. We consider the harmonic measure for a semianalytic bounded domain in the plane. We show that the har- monic measure for such a domain is analytically tame if the angles at singular boundary points are irrational multiples of . If the domain is a polygon and the angles at singular boundary points are rational or Diophantine irrational multiples of then the harmonic measure is subanalytically tame. 2000 Mathematics Subject Classification 03C64, 30C85, 32B20 (primary)
Highlights
Integration is a difficult task, both in model theory and in tame real geometry, since it is not a concept of first order
There is the following very nice result of Comte, Lion and Rolin: integrals of globally subanalytic functions with parameters with respect to the Lebesgue measure are definable in the o-minimal structure Ran,exp
We make the following definition: a measure is called subanalytically tame if there is an o-minimal expansion of Ran such that integrals of globally subanalytic functions with parameters are definable in this o-minimal expansion
Summary
Integration is a difficult task, both in model theory and in tame real geometry, since it is not a concept of first order. There is the following very nice result of Comte, Lion and Rolin (see Comte et al [4] and Lion and Rolin [26]): integrals of globally subanalytic functions with parameters with respect to the Lebesgue measure are definable in the o-minimal structure Ran,exp. We make the following definition: a measure is called subanalytically tame if there is an o-minimal expansion of Ran such that integrals of globally subanalytic functions with parameters are definable in this o-minimal expansion. In [21] it was shown, under the assumption that the angles of the domain at every singular boundary point is an irrational multiple of π , that the Dirichlet solution for a semianalytic boundary function is definable in the o-minimal structure RQ,exp (see [22] for the o-minimal structure RQ ) We use this result to show the following: Theorem A.
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