Research On Some Problems Of Non-Additive Set Functions

The theory of non-additive set functions is a new branch of mathematics. As ageneralization of the classical measure theory, non-additive set functions have beenapplied in many areas, such as knowledge engineering, artificial intelligence, gametheory, statistics, economy and sociology. Therefore, the study of non-additive setfunctions has important value of theory and application. Atoms and pseudo-atomsare studied firstly in this paper. Then we discuss the extensions of null-additive setfunctions and null-null-additive set functions on algebra of subsets. At last, we discussthe regularity of fuzzy measure on locally compact Hausdorff spaces. Our main workis as following:1. We point out that the sufficiency of Lemma 1 about null-additive set func-tions in Pap's paper (i.e. Lemma 6.4 in monograph"Null-additive Set Function") isincorrect by giving a counterexample. At the same time, we give the sufficient andnecessary condition of this problem and its proof.2. Since a lot of studies on atoms of non-additive set functions under null-additivity condition have been done, we discuss the properties of atoms of non-additive set functions under non-null-additivity condition, and obtain some results(such as Saks decomposition theorem) similar to null-additive set functions.3. We introduce the definition of pseudo-atoms of non-additive set functions andgive the relation between atoms and pseudo-atoms by the method of illustration. Wealso point out that all atoms and pseudo-atoms can be divided into three classes: classI is the set of all pseudo-atoms which are not atoms; class II is the set of all atomswhich are not pseudo-atoms; class III is the set of all atoms that are also pseudo-atoms. Specially, we point out that null-null-additivity may be an adequate framefor pseudo-atoms of non-additive set functions. We give some properties of thesepseudo-atoms which are similar to atoms of non-additive set functions as a matter ofform. Essentially, they are different. We prove a decompose theorem about atoms andpseudo-atoms under the null-additivity condition.4. We point out that the proof of Theorem 2 in Pap's paper about the extension ofnull-additive set functions on algebra of subsets is wrong by giving a counterexample and correct the old proof. We further show that this extension preserves monotonicity.Similar to the extension theorem of null-additive set functions, we give the extensiontheorem of null-null-additive set functions which satisfies monotonicity. At the sametime, we show that this extension also can keep monotonicity.5. We give the concepts of inner (outer) regular set, regular set and regular fuzzymeasure on locally compact Hausdorff space X. We obtain a necessary and sufficientcondition that a fuzzy measure is regular. At the same time, we show that the sufficientcondition that every proper difference of two compact (or compact Gδ) sets is inner(outer) regular. We show that the union of an increasing sequence of inner regularsets is inner regular, and the intersection of outer regular sets of finite measure is outerregular. For strictly monotone fuzzy measure, we show that the finite disjoint unionof inner regular sets of finite measure is inner regular and every compact (or compactGδ) set is outer regular, if and only if every bounded open set is inner regular.