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The idempotent Radon--Nikodym theorem has a converse statement

Paul Poncet 1, 2 
2 MAXPLUS - Max-plus algebras and mathematics of decision
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
Abstract : Idempotent integration is an analogue of the Lebesgue integration where $\sigma$-additive measures are replaced by $\sigma$-maxitive measures. It has proved useful in many areas of mathematics such as fuzzy set theory, optimization, idempotent analysis, large deviation theory, or extreme value theory. Existence of Radon--Nikodym derivatives, which turns out to be crucial in all of these applications, was proved by Sugeno and Murofushi. Here we show a converse statement to this idempotent version of the Radon--Nikodym theorem, i.e. we characterize the $\sigma$-maxitive measures that have the Radon--Nikodym property.
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Submitted on : Thursday, December 26, 2013 - 12:07:41 PM
Last modification on : Thursday, January 20, 2022 - 5:30:33 PM

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Paul Poncet. The idempotent Radon--Nikodym theorem has a converse statement. Information Sciences, Elsevier, 2014, pp.115-124. ⟨10.1016/j.ins.2014.02.074⟩. ⟨hal-00922377⟩



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