fuzzy measures on metric spaces abstract the aim of the present paper is to study the regularity and tightness properties of fuzzy measures on metric spaces among other results we shall give lusin s theorem and egoroff s theorem we shall also prove the singleton characterization of atom of fuzzy measure and an algorithm to evaluate the fuzzy integrals of non negative measurable functions over an atom and a union of finite disjoint atoms

lusin s theorem on fuzzy measure spaces in this paper we show that weakly null additive fuzzy measures on metric spaces possess regularity lusin s theorem which is well known in classical measure theory is generalized to fuzzy measure space by using the regularity and weakly null additivity a version of egoroff s theorem for the fuzzy measure defined on metric spaces is given an application of lusin s theorem to approximation in the mean of measurable function on fuzzy measure spaces is presented

the alexandroff theorem for riesz space valued non additive measures the alexandroff theorem for a compact non additive measure with values in a riesz space is still valid for the following two cases one is the case that the measure is autocontinuous and the riesz space has the weak asymptotic egoroff property and the other is the case that the measure is uniformly autocontinuous and the riesz space is weakly distributive a close connection between regularity and continuity of non additive measures is also given

on lusin s theorem for non additive measures that take values in an ordered topological vector space abstract lusin s theorem was established for real valued monotone measures under an equivalent condition to egoroff s theorem recently in this paper we show that the same result remains valid for non additive measures that take values in an ordered topological vector space we apply our result to several ordered topological vector spaces

on a sufficient condition of lusin s theorem for non additive measures that take values in an ordered topological vector space classical lusin s theorem is extended for real valued fuzzy measures under the weak null additivity condition recently in this paper we show similar results for non additive measures that take values in an ordered topological vector space firstly we prove lusin type theorem for weakly null additive borel measures that are continuous from above and possess an additional continuity property suggested by sun in 1994 secondly we state another one for weakly null additive fuzzy borel measures our results are applicable to several ordered topological vector spaces

the autocontinuity of set function and the fuzzy integral abstract the autocontinuity and some other concepts of a set function are introduced and sugeno s fuzzy measure with some annexed conditions is studied on a fuzzy measure space egoroffs theorem is proved it is also proved that f mean convergence is equivalent to convergence in measure for a sequence of measurable functions finally some convergence theorems of sequence of fuzzy integrals are proved especially using the concept of autocontinuity a necessary and sufficient condition for the convergence in measure theorem is given

convergence theorems for monotone measures in classical measure theory there are a number of convergence theorems such as the egorov the riesz and the lusin theorem among others we consider monotone measures i e monotone set functions vanishing in the empty set and defined on a measurable space and discuss how and to which extent classical convergence theorems can be carried over to this more general case

regularity and autocontinuity of set multifunctions in this paper we study different types of regularity for monotone uniformly autocontinuous set multifunctions defined on a ring of subsets of a locally compact hausdorff space t and taking values in the family of closed nonvoid sets of a real normed space x

lusin s theorem on monotone measure spaces in this paper a kind of regularity of monotone measures is shown by using the pseudometric generating property of set function for short p g p and an equivalence condition for egoroff s theorem lusin s theorem which is well known in classical measure theory is generalized to monotone measure spaces the continuity from above and below of set functions is not required in the discussions therefore the previous results obtained by jiang et al are generalized an approximation of measurable functions by continuous functions on monotone measure spaces is presented

on the null additivity and the autocontinuity of a fuzzy measure abstract in this paper further research on the null additivity and autocontinuity of fuzzy measures is made we prove that autocontinuity from below and autocontinuity from above of a fuzzy measure are equivalent when this fuzzy measure is finite furthermore if the space x is countable then the null additivity is equivalent to the autocontinuity

a lusin type theorem for regular monotone uniformly autocontinuous set multifunctions in this paper we continue our study concerning monotone uniformly autocontinuous regular set multifunctions as an application a lusin type theorem under a suitable type of measurability is obtained and several applications for finitely purely atomic multisubmeasures are proved

lusin s theorem for monotone set valued measures on topological spaces abstract on a topological space the inner regularity outer regularity and regularity of a monotone set valued measure are considered their properties are discussed based on these the versions of egoroff s theorem and lusin s theorem for monotone set valued measures on a topological space are shown respectively

asymptotic structural characteristics of fuzzy measure and their applications abstract in fuzzy measure theory as sugeno s fuzzy measures lose additivity in general the concept almost which is well known in classical measure theory splits into two different concepts almost and pseudo almost in order to replace the additivity it is quite necessary to investigate some asymptotic behaviors of a fuzzy measure at sequences of sets which are called waxing and waning and to introduce some new concepts such as autocontinuity converse autocontinuity and pseudo autocontinuity these concepts describe some asymptotic structural characteristics of a fuzzy measure in this paper by means of the asymptotic structural characteristics of fuzzy measure we also give four forms of generalization for both egoroff s theorem riesz s theorem and lebesgue s theorem respectively and prove the almost everywhere pseudo almost everywhere convergence theorem the convergence in measure pseudo in measure theorem of the sequence of fuzzy integrals in the last two theorems the employed conditions are not only sufficient but also necessary

fuzzy regular measures on topological spaces in this paper the concepts of the inner regularity outer regularity and regularity of a fuzzy measure are introduced on a topological space their properties are studied in detail egoroff theorem and lusin theorem are proved finally the conditions for a fuzzy measure to be tight and perfect are given respectively

atoms of weakly null additive monotone measures and integrals in this paper we prove some properties of atoms of weakly null additive monotone measures by using the regularity and weak null additivity a sin gleton characterization of atoms of monotone measures on a metric space is shown it is a generalization of previous results obtained by pap the calculation of the sugeno integral and the choquet integral over an atom is also presented respectively similar results for recently introduced universal integral are also given following these results it is shown that the sugeno integral and the choquet integral over an atom of monotone measure is maxitive linear and standard linear respectively convergence theorems for the sugeno integral and the choquet integral over an atom of a monotone measure are also shown

mirror duality in a joyce manifold abstract previously the two of the authors defined a notion of dual calabi yau manifolds in a g 2 manifold and described a process to obtain them here we apply this process to a compact g 2 manifold constructed by joyce and as a result we obtain a pair of borcea voisin calabi yau manifolds which are known to be mirror duals of each other

semi global exponential stabilization of linear systems subject to input saturation via linear feedbacks abstract it is known that a linear time invariant system subject to input saturation can be globally asymptotically stabilized if it has no eigenvalues with positive real parts it is also shown by fuller 1997 and sussmann and yang 1991 that in general one must use nonlinear control laws and only some special cases can be handled by linear control laws in this paper we show the existence of linear state feedback and or output feedback control laws for semi global exponential stabilization rather than global asymptotic stabilization of such systems we explicitly construct linear static state feedback laws and or linear dynamic output feedback laws that semi globally exponentially stabilize the given system our results complement the negative result of fuller 1977 and sussmann and yang 1991

on the regularity of the fuzzy measure on metric fuzzy measure spaces abstract in this paper the concept of the regular fuzzy measure on metric spaces is introduced some of its properties are discussed and lusin s theorem is shown in addition a condition for which a uniformly autocontinuous set function is a fuzzy measure is given

a parametric lyapunov equation approach to the design of low gain feedback low gain feedback has found several applications in constrained control systems robust control and nonlinear control low gain feedback refers to a family of stabilizing state feedback gains that are parameterized in a scalar and go to zero as the scalar decreases to zero such feedback gains can be constructed either by an eigenstructure assignment algorithm or through the solution of a parametric algebraic riccati equation are the eigenstructure assignment approach leads to feedback gains in the form of a matrix polynomial in the parameter while the are approach requires the solution of an are for each value of the parameter this note proposes an alternative approach to low gain feedback design based on the solution of a parametric lyapunov equation such an approach possesses the advantages of both the eigenstructure assignment approach and the are based approach it also avoids the possible numerical stiffness in solving a parametric are and the structural decomposition of the open loop system that is required by the eigenstructure assignment approach

some notes on the regularity of fuzzy measures on metric spaces abstract the purpose of this paper is to correct and modify some conclusions shown in a paper by wu congxin and ha minghu fuzzy sets and systems 66 1994 373 379 and to give some further results

set valued lusin type theorem for null null additive set multifunctions in a previous paper we presented a set valued egoroff type theorem for monotone set multifunctions in this paper a new version of this set valued egoroff type theorem is obtained it enables a set valued lusin type theorem for null null additive monotone continuous set multifunctions to be established several applications of this lusin type theorem are given pointing out some relationships among m measurability a measurability and totally measurability of real functions we also compare our new lusin type theorem with lusin s theorem previously obtained by the second author

semi global exponential stabilization of linear discrete time systems subject to input saturation via linear feedbacks abstract in this paper we show that a linear discrete time system subject to input saturation is semi globally exponentially stabilizable via linear state and or output feedback laws as long as the system in the absence of input saturation is stabilizable and detectable and has all its poles located inside or on the unit circle furthermore the semi globally stabilizing feedback laws are explicitly constructed the results presented here are parallel to our earlier results on the continuous time counterpart lin and saberi 1993