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Intégration des fonctions sous-analytiques et volumes des sous-ensembles sous-analytiques. (Integration of subanalytic functions and volumes of subanalytic subspaces). (French) Zbl 0912.32007

The work speaks about the integration of global subanalytic functions and seems very useful for the calculus. As the authors say, it was inspired by L. van den Dries and Rémi Langevin, both outstanding mathematicians with many great ideas. The proofs use in a very interesting way the preparation theorem by the authors [J.-M. Lion and J.-P. Rolin, Ann. Inst. Fourier 47, No. 3, 859-884 (1997; Zbl 0873.32004)] and a very useful, unfortunately not often used, work by K. Kurdyka and G. Raby [Ann. Inst. Fourier 39, No. 3, 753-771 (1989; Zbl 0673.32015)] about the density of subanalytic sets.
It is a pity that the Introduction (I) is dispersed and misleading, with no order in the ideas, quotations and an overly optimistic statement about the possibility of obtention of algebras stable by integration. The reviewer is tempted to advise a beginner reader to skip the introduction and read the text first, as well as not to be impressed by the bibliography, most of which is not used except in the introduction.

MSC:

32B20 Semi-analytic sets, subanalytic sets, and generalizations
32B15 Analytic subsets of affine space
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References:

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