6,095 research outputs found
Autocontinuity and convergence theorems for the Choquet integral
Our aim is to provide some convergence theorems for the Choquet integral with respect to various notions of convergence. For instance, the dominated convergence theorem for almost uniform convergence is related to autocontinuous set functions. Autocontinuity can also be related to convergence in measure, strict convergence or mean convergence. Whereas the monotone convergence theorem for almost uniform convergence is related to monotone autocontinuity, a weaker version than autocontinuity.
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
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
Commutative POVMs and Fuzzy Observables
In this paper we review some properties of fuzzy observables, mainly as
realized by commutative positive operator valued measures. In this context we
discuss two representation theorems for commutative positive operator valued
measures in terms of projection valued measures and describe, in some detail,
the general notion of fuzzification. We also make some related observations on
joint measurements.Comment: Contribution to the Pekka Lahti Festschrif
From probability to sequences and back
This is a survey covering sequential structures and their
applications to the foundations of probability theory. Sequential convergence, convergence groups and the extension of sequentially continuous
maps belong to general topology and Trieste for long has been a center
of sequential topology. We begin with some personal reflections, con-
tinue with topological problems motivated by the extension of probability
measures, and close with some recent results related to the categorical
foundations of probability theory
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