95 research outputs found
The idempotent Radon--Nikodym theorem has a converse statement
Idempotent integration is an analogue of the Lebesgue integration where
-additive measures are replaced by -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 -maxitive measures that have the
Radon--Nikodym property.Comment: 13 page
Representation of maxitive measures: an overview
Idempotent integration is an analogue of Lebesgue integration where
-maxitive measures replace -additive measures. In addition to
reviewing and unifying several Radon--Nikodym like theorems proven in the
literature for the idempotent integral, we also prove new results of the same
kind.Comment: 40 page
The Choquet integral as Lebesgue integral and related inequalities
summary:The integral inequalities known for the Lebesgue integral are discussed in the framework of the Choquet integral. While the Jensen inequality was known to be valid for the Choquet integral without any additional constraints, this is not more true for the Cauchy, Minkowski, Hölder and other inequalities. For a fixed monotone measure, constraints on the involved functions sufficient to guarantee the validity of the discussed inequalities are given. Moreover, the comonotonicity of the considered functions is shown to be a sufficient constraint ensuring the validity of all discussed inequalities for the Choquet integral, independently of the underlying monotone measure
On the Minkowski-H\"{o}lder type inequalities for generalized Sugeno integrals with an application
In this paper, we use a new method to obtain the necessary and sufficient
condition guaranteeing the validity of the Minkowski-H\"{o}lder type inequality
for the generalized upper Sugeno integral in the case of functions belonging to
a wider class than the comonotone functions. As a by-product, we show that the
Minkowski type inequality for seminormed fuzzy integral presented by Daraby and
Ghadimi in General Minkowski type and related inequalities for seminormed fuzzy
integrals, Sahand Communications in Mathematical Analysis 1 (2014) 9--20 is not
true. Next, we study the Minkowski-H\"{o}lder inequality for the lower Sugeno
integral and the class of -subadditive functions introduced in On
Chebyshev type inequalities for generalized Sugeno integrals, Fuzzy Sets and
Systems 244 (2014) 51--62. The results are applied to derive new metrics on the
space of measurable functions in the setting of nonadditive measure theory. We
also give a partial answer to the open problem 2.22 posed by
Borzov\'a-Moln\'arov\'a and et al in The smallest semicopula-based universal
integrals I: Properties and characterizations, Fuzzy Sets and Systems 271
(2015) 1--17.Comment: 19 page
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