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

a note on the null additivity of the fuzzy measure corrigendum to on the null additivity of the fuzzy measure fuzzy sets and systems 78 1996 337 339 in this paper we point out that theorems 2 and 2 shown in the paper on the null additivity of the fuzzy measure fuzzy sets and systems 78 1996 337 339 are not valid and we also introduce a weaker condition than the null additivity of a set function to correct the results

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

a possibility measure is not a fuzzy measure abstract in this note we show that a possibility measure is not a particular type of fuzzy measure except in trivial cases

lebesgue theorems in non additive measure theory in this paper the sufficient and necessary conditions for different kinds of lebesgue theorem in non additive measure theory are presented respectively the equivalence among the lebesgue theorem the monotone convergence theorems of fuzzy and of choquet integral are shown as direct results of the lebesgue theorems the necessary conditions for egoroff s theorem are given

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

the choquet integral in riesz space a comprehensive discussion of the theory of choquet integration in a riesz space is given in particular it is proved that the monotone convergence theorem the fatou lemma and the dominated convergence theorem are still valid for riesz space valued non additive measures if we assume that the riesz space has a new property concerning the cardinality of the set of points of discontinuity of a monotone function

autocontinuity and convergence theorems for the choquet integral we provide convergence theorems for the choquet integral for various notions of convergence monotone almost uniform convergence from above below is associated to monotone autocontinuity from above below the dominated almost uniform convergence is associated to monotone autocontinuity autocontinuity can be associated to the dominated convergence in measure in mean or in strict measure

duality and ordinality in fuzzy measure theory this paper reformulates the duality in fuzzy measure theory discussed by liu as the cardinal duality principle this principle guarantees that if a proposition concerning a finite fuzzy measure holds then its dual also holds it is however not applicable to propositions concerning infinite fuzzy measures by introducing the concept of ordinality the author converts the principle into the ordinal duality principle which is applicable to all ordinal propositions concerning general finite or infinite fuzzy measures

regularity aspects of non additive set multifunctions abstract in this paper continuity properties of set multifunctions such as regularity and continuity from above and below as well as others are introduced by way of a wijsman topology some of the relationships between them are established different examples counterexamples and applications are provided

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 the comonotonic property for sugeno integral the so called comonotonic property for sugeno integral is introduced and investigated where 0 2 0 is a binary operation under the assumption that lim b a email protected a and lim a a email protected b we prove that sugeno integral possesses comonotonic property if and only if equals to one of the four operators minimum maximum pf the first projection and pl the last projection we also discuss comonotonic property when this assumption does not hold

on sufficient condition for the egoroff theorem of an ordered vector space valued non additive measure in this paper we show that the egoroff theorem remains valid for any ordered vector space valued non additive measure if the measure is multiple continuous from above and continuous from below

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

relationship among continuity conditions and null additivity conditions in non additive measure theory the paper shows all the implication relationship among six continuity conditions and two null additivity conditions with respect to non additive measures the six continuity conditions are continuity from above order continuity strong order continuity exhaustivity continuity from below and null continuity and the two null additivity conditions are null additivity and weak null additivity the implication relationship is summarized in a diagram

on egoroff s theorems on fuzzy measure spaces in this paper we show that egoroff s theorem which is well known in classical measure theory remains valid for a fuzzy measure without any additional condition previous results obtained by wang zhenyuan and others are improved egoroff s theorem on fuzzy measure spaces is formulated in full generality taylor s theorem which concerns convergence almost everywhere in classical measure theory is also extended to fuzzy measure spaces

continuity and compactness of the indirect product of two non additive measures let x and y be non empty sets e a field of subsets of x and f a field of subsets of y then every uniformly autocontinuous indirect product of two non additive measures m on e and n on f is continuous on the product field generated by e and f whenever m is continuous and n is compact a similar result holds for the compactness of indirect product measures

regularity and lusin s theorem for riesz space valued fuzzy measures a smoothness condition the multiple egoroff property is introduced and imposed on a riesz space to show that every weakly null additive riesz space valued fuzzy borel measure on any metric space is regular it is also proved that lusin s theorem remains valid for such riesz space valued non additive measures

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

on sufficient conditions for the egoroff theorem of an ordered topological vector space valued non additive measure in this paper we consider an ordered vector space endowed with a locally full topology which is called an ordered topological vector space we show that the egoroff theorem remains valid for the ordered topological vector space valued non additive measure in the following four cases the first case is that the measure is strongly order totally continuous the second case is that the measure is strongly order continuous and possesses an additional continuity property suggested by sun in 1994 when the ordered topological vector space has a certain property the third case is that the measure is continuous from above and below when the topology is locally convex the fourth case is that the measure is uniformly autocontinuous from above continuous from below and strongly order continuous when the topology is locally convex our results are applicable to several ordered topological vector spaces

convergence of sequence of measurable functions on fuzzy measure spaces abstract the purpose of this paper is to investigate the convergence of sequence of measurable functions on fuzzy measure spaces several classical results on the convergence of measurable functions such as egoroff s theorem lebesgue s theorem and riesz s theorem are extended to fuzzy measure spaces by using the pseudometric generating property and order continuity of fuzzy measures

order continuous of monotone set function and convergence of measurable functions sequence in this paper we introduce the concept of strongly order continuity of a set function and use it to investigate the convergence of measurable functions sequence and of integral sequence on fuzzy measure space several equivalent conditions of strongly order continuity are presented it is shown that a necessary and sufficient condition that lebesgue s theorem remains valid for a monotone set function is that the set function possesses strongly order continuity

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

a new necessary and sufficient condition for the egoroff theorem in non additive measure theory abstract this paper shows that a newly defined condition called condition m is a necessary and sufficient condition for the egoroff theorem in non additive measure theory the existing necessary and sufficient conditions for the egoroff theorem are described by a doubly indexed sequence of measurable sets while condition m is described by a singly indexed sequence of measurable sets

property s of fuzzy measure and riesz s theorem abstract in this note a sufficient and necessary condition for riesz s theorem to hold in the fuzzy measure space is given

pseudo atoms and darboux property for set multifunctions in this paper we introduce and study the concepts of pseudo atom and darboux property for set multifunctions we establish relationships between the darboux property and non atomicity in the multivalued case we also prove that the range of a multimeasure having the darboux property is a convex set

conditions for egoroff s theorem in non additive measure theory abstract this paper gives necessary and or sufficient conditions for egoroff s theorem in non additive measure theory a necessary and sufficient condition described without measurable functions two sufficient conditions and a necessary condition one of the two sufficient conditions is strong order total continuity continuity at measurable sets of measure zero with respect to net convergence and the other is strong order continuity sequential continuity at measurable sets of measure zero together with property s the necessary condition is strong order continuity in addition the paper shows the following continuity from above and below which is a known sufficient condition and the above mentioned two sufficient conditions are independent of each other the disjunction of these three sufficient conditions is not a necessary condition if the underlying set is at most countable then strong order continuity is necessary and sufficient and generally strong order continuity is not sufficient

on egoroff s theorems on finite monotone non additive measure space in this note we give four versions of egoroff s theorem in non additive measure theory by using condition e the pseudo condition e of set function and the duality relations between the conditions these conditions offered are not only sufficient but also necessary for the four kinds of egoroff s theorems

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

algorithms for computing a parameterized st orientation st orientations st numberings or bipolar orientations of undirected graphs are central to many graph algorithms and applications several algorithms have been proposed in the past to compute an st orientation of a biconnected graph in this paper we present new algorithms that compute such orientations with certain parameterized characteristics in the final st oriented graph such as the length of the longest path this work has many applications including graph drawing and network routing where the length of the longest path is vital in deciding certain features of the final solution this work applies to other difficult problems as well such as graph coloring and of course longest path we present extended theoretical and experimental results which show that our technique is efficient and performs well in practice

autocontinuity convergence in measure and convergence in distribution abstract it is shown that the autocontinuity is equivalent to the property that the convergence in measure implies the convergence in distribution this result is applied to denneberg s dominated convergence theorem for the choquet integral

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

factoring integers with elliptic curves this paper is devoted to the description and analysis of a new algorithm to factor positive integers it depends on the use of elliptic curves the new method is obtained from pollard s p 1 method proc cambridge philos soc 76 1974 521 528 by replacing the multiplicative group by the group of points on a random elliptic curve it is conjectured that the algorithm determines a non trivial divisor of a composite number n in expected time at most k p log n 2 where p is the least prime dividing n and k is a function for which log k x 2 o 1 log x log log x for x x o in the worst case when n is the product of two primes of the same order of magnitude this is

generalized convergence theorems for monotone measures abstract in this paper we propose three types of absolute continuity for monotone measures and present some of their basic properties by means of these three types of absolute continuity we establish generalized egoroff s theorem generalized riesz s theorem and generalized lebesgue s theorem in the framework involving the ordered pair of monotone measures the egoroff theorem the riesz theorem and the lebesgue theorem in the traditional sense concerning a unique monotone measure are extended to the general case these three generalized convergence theorems include as special cases several previous versions of egoroff like theorem riesz like theorem and lebesgue like theorem for monotone measures 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

lattice valued fuzzy measure and lattice valued fuzzy integral abstract in this paper 1 the concepts of lattice valued fuzzy measure with no valuation property and lower resp upper lattice valued fuzzy integral are proposed which give the unified discription to the fuzzy measures and fuzzy integrals studied by delgado and moral qiao ralescu adams sugeno wang and zhang 2 some asymptotic structural characteristics of lattice valued fuzzy measures are introduced and some relations between them are given 3 some concepts of convergences for lattice valued functions are defined and riesz theorem egoroff s theorems and lebesgue s theorem for lattice valued measurable function are proved 4 the monotone increasing resp decreasing convergence theorem and almost resp pseudo almost everywhere convergence theorrem for lower resp upper lattice valued fuzzy integral are shown under some weak conditions

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

regularities of riesz space valued non additive measures with applications to convergence theorems for choquet integrals a deeper investigation of radonness and t smoothness properties of riesz space valued borel non additive measures is carried out to this end due to a lack of e argument in a riesz space the multiple egoroff property is introduced and enforced on the involved riesz space the established regularity properties of borel non additive measures are instrumental when formulating certain types of monotone convergence theorems for choquet integrals

the egoroff property and the egoroff theorem in riesz space valued non additive measure theory the egoroff theorem remains valid for any riesz space valued non additive measure which is strong order continuous and possesses a form of continuity called property s in the literature whenever the riesz space has the egoroff property this version of the egoroff theorem is also valid for any non additive measure with the property of uniform autocontinuity strong order continuity and continuity from below by assuming only the weak s distributivity which is weaker than the egoroff property

the continuity and compactness of riesz space valued indirect product measures the purpose of the paper is to extend the classical marczewski and ryll nardzewski theorems for the continuity and compactness of measures to riesz space valued non additive measures

a further investigation for fuzzy measures on metric spaces the pseudometric generating property of non additive set functions also plays an important role as the autocontinuity in fuzzy measure theory jiang ph d thesis chiba university 1996 wang and klir fuzzy measure theory plenum press new york 1992 a further research on this property has been presented in jiang et al fuzzy sets and systems 1998 the aim of this paper is to prove the egoroff s theorem and lusin s theorem for measurable functions on metric spaces hold for those fuzzy measures with the exhaustivity and pseudometric generating property

general chebyshev type inequalities for sugeno integrals chebyshev type inequalities for the sugeno integral on abstract spaces are studied in a rather general form thus closing the series of papers on the topic dealing with special cases restricted to the real line and product operation

compact visibility representation of 4 connected plane graphs the visibility representation vr for short is a classical representation of plane graphs the vr has various applications and has been extensively studied in the literature a main focus of the study is to minimize the size of the vr it is known that there exists a plane graph g with n vertices where any vr of g requires a size at least 2n3 x 4n3 3 for upper bounds it is known that every plane graph has a vr with size at most 23n x 2n 5 and a vr with size at most n 1 x 43n it has been an open problem to find a vr with both height and width simultaneously bounded away from the trivial upper bounds namely of size c hnxc wn with c h 1 and c w 2 in this paper we provide the first vr construction for a non trivial graph class that simultaneously bounds both the height and the width we prove that every 4 connected plane graph has a vr with height 3n4 2 n 4 and width 3n2 our vr algorithm is based on an st orientation of 4 connected plane graphs with special properties the area of the vr presented in this paper is larger than the area of some of the previous results for this graph class however bounding one dimension of the vr only requires finding a good st orientation or a good dual s t orientation of g on the other hand bounding both dimensions of the vr requires finding a good st orientation and a good dual s t orientation of g at the same time and hence is far more challenging since the st orientation is a very useful concept in other applications this result may be of independent interest reducing the height the width respectively of the vr is the same as reducing the length of the longest path in an st orientation of g dual st orientation respectively we show that it s np complete to find an st orientation of a 2 connected plane graph that minimizes the sum of the length of the longest path in the orientation and the length of the longest path in its dual orientation

on the convergence of measurable set valued function sequence on fuzzy measure space in this paper we first discuss the measurable projection theorem on fuzzy measure space and in this framework the characterization theorem with respect to measurability of a set valued function is given by means of the asymptotic structural characteristics of fuzzy measure we discuss four forms of generalization for both lebesgue s theorem riesz s theorem and egoroff s theorem respectively the relation between convergence of measurable set valued function sequence and that of corresponding measurable real valued function sequence are also discussed

a jensen type inequality for fuzzy integrals in this paper we show a jensen type inequality for the sugeno integral we also discuss some conditions assuring the satisfaction of opposite inequality reverse jensen inequality

on the regularities of fuzzy set multifunctions with applications in variation extensions and fuzzy set valued integrability problems in this paper we continue our studies 6 7 concerning regularity in this sense we obtain extension results by preserving regularity and we establish basic properties and continuity properties for the variation of fuzzy set multifunctions this enables us to get relationships among different types of regularity and also to point out regularity properties of the fuzzy set valued integral 20

the first case of fermat s last theorem is true for all prime exponents up to 714 591 416 091 389 we show that if the first case of fermat s last theorem is false for prime exponent p then p2 divides qp q for all primes q sq as a corollary we state the theorem of the title 1988 american mathematical society

convergence of a sequence of fuzzy number valued fuzzy measurable functions on the fuzzy number valued fuzzy measure space abstract in this paper some new concepts convergences of a sequence of fuzzy number valued fuzzy measurable functions on the fuzzy set are introduced and the relations between these convergences are discussed on the fuzzy number valued fuzzy measure space riesz s theorem egoroff s theorem and lebesgue s theorem are proved

fuzzy measures and integrals we present fuzzy measures and fuzzy integrals as special poset homeomorphisms besides general fuzzy integrals regular fuzzy integrals are introduced and some of their properties are discussed state of art of fuzzy measure and fuzzy integral theory are briefly summarized and some further streaming is sketched

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 set valued egoroff type theorem in this paper an egoroff type theorem for a fuzzy multimeasure defined on a s algebra of subsets of an abstract space and taking values in the family of all nonvoid closed subsets of a real normed space is established some necessary and or sufficient conditions under which egoroff s theorem is valid are also presented

inner and outer representation by choquet integral the representations of comonotonically additive and monotone functionals are investigated an i regular non additive measure is defined as a non additive measure whose values can be calculated from its values on the inner compact sets and the values on compact sets can be calculated from its values on the outer open sets the basic properties of an i regular non additive measure are shown the inner representation of the functional is proposed and is compared with the outer representation

non atomicity for fuzzy and non fuzzy multivalued set functions in this paper we extend the concept of set valued fuzzy measures introduced and studied by guo and zhang 2004 on set valued fuzzy measures information sciences 160 2004 13 25 to the class of multivalued set functions taking values in the family of non empty closed subsets of a real normed space also we introduce the notions of atom and non atomicity of a multivalued set function and present some of their properties we also establish an extension result by preserving non atomicity for a fuzzy multisubmeasure multimeasure

the egoroff theorem for non additive measures in riesz spaces the egoroff theorem remains valid for any riesz space valued non additive measure which is continuous from above and below by assuming that the riesz space has the asymptotic egoroff property this property is satisfied for many concrete riesz spaces such as rs of all real functions on a non empty set s l0 0 1 of all lebesgue measurable functions and their ideals

algorithm to find a maximum 2 packing set in a cactus abstract in this paper we address the problem of finding a maximum 2 packing set in a cactus let g v g e g be an undirected connected graph then a subset s v g is a 2 packing set if for every pair of vertices u v s the shortest path between them is at least three edges long this type of sets results in a wide range of applications such as the study of molecules network modeling allocation of facilities and frequency assignment the cactus is a planar graph such that any edge belongs to at most one cycle hitherto to the best of the authors knowledge no algorithm finds a maximum 2 packing set in a cactus this problem is np hard in arbitrary graphs our approach to solving this problem was first finding a maximum 2 packing set in a unicyclic graph then we adapted this algorithm to run in a cactus the time complexity of the resulting algorithm is o n 2 where n v g

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

autocontinuity of set valued fuzzy measures and applications the main object of this paper is to further the study of set valued fuzzy measures introduced by gavrilut in 2009 some asymptotic structural characteristics of set valued fuzzy measures such as null additivity and autocontinuity are investigated in terms of applications convergence theorems for sequences of measurable functions are established

np completeness of st orientations for plane graphs an st orientation or bipolar orientation of a 2 connected graph g is an orientation of its edges to generate a directed acyclic graph with a single source s and a single sink t given a plane graph g and two exterior vertices s and t the problem of finding an optimum st orientation i e an st orientation in which the length of a longest st path is minimized was first proposed indirectly by rosenstiehl and tarjan in and then later directly by he and kao in in this paper we prove that given a 2 connected plane graph g two exterior vertices s t and a positive integer k the decision problem of whether g has an st orientation where the maximum length of an st path is k is np complete this solves a long standing open problem on the complexity of optimum st orientations for plane graphs as a by product we prove that the np completeness result holds for planar graphs as well

a theory of fuzzy measures representations the choquet integral and null sets abstract some theorems of t murofushi and m sugeno fuzzy sets and systems 29 1989 201 227 concerning representation of fuzzy measures and the choquet integral are generalized it is shown that if there is a certain relation between two measurable functions then the choquet integral is additive for these two functions in addition this article discusses null sets with respect to fuzzy measures and also fuzzy measures defined on a class which is not a algebra