Stability under integration of sums of products of real globally subanalytic functions and their logarithms

Cluckers, Raf; Miller, Daniel J.

doi:10.1215/00127094-2010-213

Cited by 29 publications

(48 citation statements)

References 21 publications

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“…By Proposition 6.4(2) there is some h ∈ P >0 such that |g(x) − S h g(x)| < ε for all x ∈ P. Let v := S h g. Then v is P-constructible by Proposition 4. 10. By Proposition 6.4(1) we have that v is C ∞ .…”

Section: Proofmentioning

confidence: 83%

“…Then F is a P-constructible function by Proposition 4. 10. By the parametric version of Construction 2.6 and by applying the familiar transfer argument, we get that lim s→∞,s∈P f s dλ P,n = lim s→∞,s∈S F S (s).…”

Section: Lebesgue's Theorem On Dominated Convergence and The Fundamenmentioning

confidence: 95%

“…It is also shown that the set Fin k (A) is semialgebraic if A is semialgebraic. A function is called constructible if it is a finite sum of finite products of globally subanalytic functions and logarithms of positive globally subanalytic functions (see Cluckers and Miller [10][11][12]). We need the following version of the above theorem (where λ n denotes the Lebesgue measure on R n ):…”

Section: Integration Of Parameterized Definable Functionsmentioning

confidence: 99%

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Lebesgue measure and integration theory on non-archimedean real closed fields with archimedean value group

Kaiser

2017

Proc. London Math. Soc.

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Abstract. Given a non-archimedean real closed field with archimedean value group which contains the reals, we establish for the category of semialgebraic sets and functions a full Lebesgue measure and integration theory such that the main results from the classical setting hold. The construction involves methods from model theory, ominimal geometry and valuation theory. We set up the construction in such a way that it is determined by a section of the valuation. If the value group is isomorphic to the group of rational numbers the construction is uniquely determined up to isomorphism. The range of the measure and integration is obtained in a controlled and tame way from the real closed field we start with. The main example is given by the case of the field of Puiseux series where the range is the polynomial ring in one variable over this field.

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Section: Proofmentioning

confidence: 83%

Section: Lebesgue's Theorem On Dominated Convergence and The Fundamenmentioning

confidence: 95%

Section: Integration Of Parameterized Definable Functionsmentioning

confidence: 99%

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Lebesgue measure and integration theory on non-archimedean real closed fields with archimedean value group

Kaiser

2017

Proc. London Math. Soc.

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“…By the usual transfer argument we obtain semialgebraic versions of the transformation formula and of Lebesgue's theorem on dominated convergence. We present the latter: For the fundamental theorem of calculus and Fubini's theorem we use the results of Cluckers and D. Miller [25,26,27] who have extended the work of Comte, Lion and Rolin. We formulate the latter result.…”

Section: Main Results Of Integrationmentioning

confidence: 99%

Mini-Workshop: Surreal Numbers, Surreal Analysis, Hahn Fields and Derivations

2017

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New striking analogies between H. Hahn's fields of generalised series with real coefficients, G. H. Hardy's field of germs of real valued functions, and J. H. Conway's field No of surreal numbers, have been lately discovered and exploited. The aim of the workshop was to bring quickly together experts and young researchers, to articulate and investigate current key questions and conjectures regarding these fields, and to explore emerging applications of this recent discovery.

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“…However, by the seminal work of Comte, Lion and Rolin [5,17] (see also [11]), later extended by Cluckers and D. Miller [1,2,3], it is enough to add the logarithm to do integration on R an . Hence the existence of a logarithm is essential in this regard.…”

mentioning

confidence: 99%

Logarithms, constructible functions and integration on non-archimedean models of the theory of the real field with restricted analytic functions with value group of finite archimedean rank

Kaiser

2022

Fund. Math.

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Given a model of the theory of the real field with restricted analytic functions such that its value group has finite archimedean rank, we show how one can extend the restricted logarithm to a global logarithm with values in the polynomial ring over the model with dimension the archimedean rank. The logarithms are determined by algebraic data from the model, namely by a section of the model and by an embedding of the value group into its Hahn group. If the archimedean rank of the value group coincides with the rational rank, the logarithms are equivalent. We illustrate how one can embed such a logarithm into a model of the real field with restricted analytic functions and exponentiation. This allows us to define constructible functions with good lifting properties. As an application we establish a Lebesgue measure and integration theory with values in the polynomial ring, extending and strengthening the construction in [T. Kaiser, Proc.

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Stability under integration of sums of products of real globally subanalytic functions and their logarithms

Cited by 29 publications

References 21 publications

Lebesgue measure and integration theory on non-archimedean real closed fields with archimedean value group

Lebesgue measure and integration theory on non-archimedean real closed fields with archimedean value group

Mini-Workshop: Surreal Numbers, Surreal Analysis, Hahn Fields and Derivations

Logarithms, constructible functions and integration on non-archimedean models of the theory of the real field with restricted analytic functions with value group of finite archimedean rank

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