1,802 research outputs found
How regular can maxitive measures be?
We examine domain-valued maxitive measures defined on the Borel subsets of a
topological space. Several characterizations of regularity of maxitive measures
are proved, depending on the structure of the topological space. Since every
regular maxitive measure is completely maxitive, this yields sufficient
conditions for the existence of a cardinal density. We also show that every
outer-continuous maxitive measure can be decomposed as the supremum of a
regular maxitive measure and a maxitive measure that vanishes on compact
subsets under appropriate conditions.Comment: 24 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
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
Generalized closed sets in ditopological texture spaces with application in rough set theory
In this paper, the counterparts of generalized open (g-open) and generalized closed (g-closed) sets for ditopological texture spaces are introduced and some of their characterizations are obtained. Some characterizations are presented for generalized bicontinuous difunctions. Also, we introduce new notions of compactness and stability in ditopological texture spaces based on the notion of g-open and g-closed sets. Finally, as an application of g-open and g-closed sets, we generalize the subsystem based denition of rough set theory by using new subsystem, called generalized open sets to dene new types of lower and upper approximation operators, called g-lower and g-upper approximations. These decrease the upper approximation and increase the lower approximation and hence increase the accuracy. Properties of these approximations are studied. An example of multi-valued information systems are given
Inclusion hyperspaces and capacities on Tychonoff spaces: functors and monads
The inclusion hyperspace functor, the capacity functor and monads for these
functors have been extended from the category of compact Hausdorff spaces to
the category of Tychonoff spaces. Properties of spaces and maps of inclusion
hyperspaces and capacities (non-additive measures) on Tychonoff spaces are
investigated
Expansiveness, Lyapunov exponents and entropy for set valued maps
In this paper we introduce a notion of expansiveness for a set valued map
defined on a topological space different from that given by Richard Williams at
\cite{Wi, Wi2} and prove that the topological entropy of an expansive set
valued map defined on a Peano space of positive dimension is greater than zero.
We define Lyapunov exponent for set valued maps and prove that positiveness of
its Lyapunov exponent implies positiveness for the topological entropy. Finally
we introduce the definition of (Lyapunov) stable points for set valued maps and
prove a dichotomy for the set of stable points for set valued maps defined on
Peano spaces: either it is empty or the whole space.Comment: 24 pages, 1 figur
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