The purpose of this paper is to investigate the integrability, measurability, and summability of certain set functions. The paper is divided into four chapters. The first chapter contains basic definitions and preliminary remarks about set functions and absolute continuity. In Chapter i, the integrability of bounded set functions is investigated. The chapter culminates with a theorem that characterizes the transmission of the integrability of a real function of n bounded set functions. In Chapter III, measurability is defined and a characterization of the transmission of measurability by a function of n variables is provided, In Chapter IV, summability is defined and the summability of set functions is investigated, Included is a characterization of the transmission of summability by a function of n variables.

by Kam Siu Man. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1991. / Includes bibliographical references. / Chapter Chapter I --- On a generalized Kuratowski's closure theorem / Chapter §0. --- Introduction --- p.1 / Chapter §1. --- Counter examples and the idea of migration --- p.1 / Chapter §2. --- The use of immigration regulations --- p.11 / Chapter §3. --- Remarks on the boundary function and the complementary function --- p.35 / Chapter Chapter II --- A problem in number theory / Chapter §1. --- Introduction --- p.37 / Chapter §2. --- Preliminaries --- p.37 / Chapter §3. --- Technical Lemmas --- p.41 / Chapter §4. --- Main Theorems --- p.45 / Chapter §5. --- Closing Remarks --- p.49 / Chapter Chapter III --- Stiff elements in semigroups and their expansions / Chapter §1. --- Introduction --- p.51 / Chapter §2. --- String expansions of semigroups --- p.51 / Chapter §3. --- Stiff Semigroups --- p.59 / Chapter §4. --- Construction of examples --- p.71 / Chapter §5. --- Partition of semigroups by stiffness --- p.74

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In leading up to the proof, methods for constructing fields and finitely additive set functions are introduced with an application involving the Tagaki function given as an example. Also, non-absolutely continuous set functions are constructed using Banach limits and maximal filters.

The investigation of this thesis is introduced by finding all the relationships existing among the various types of set functions. Characterizations of the different set functions will be formulated in order to facilitate the construction and identification of each type of set function considered. This problem has important applications to measure theory and integration.Ball State UniversityMuncie, IN 47306

Thesis (Ph.D)--Mathematics, Georgia Institute of Technology, 2009. / Committee Chair: Croot, Ernie; Committee Member: Costello, Kevin; Committee Member: Lyall, Neil; Committee Member: Tetali, Prasad; Committee Member: Yu, XingXing. Part of the SMARTech Electronic Thesis and Dissertation Collection.

Settings and notions are as in previous abstracts of W. D. L. Appling. This paper contains an investigation of the relationship between a class of non-linear functions defined on PAB and certain subspaces of PAB in particular Appling's linear C-sets, Solomon leader's finitely additive Lp spaces, and one of the projective limit spaces studied by Davis, Murray, and Weber.

The first chapter gives basic definitions and theorems concerning set functions and set function integrals. The lemmas and theorems are presented without proof in this chapter. The second chapter deals with absolute continuity and Lipschitz condition. Particular emphasis is placed on the properties of max and min integrals. The third chapter deals with approximating absolutely continuous functions with bounded functions. It also deals with the existence of the integrals composed of various combinations of bounded functions and finitely additive functions. The concluding theorem states if the integral of the product of a bounded function and a non-negative finitely additive function exists, then the integral of the product of the bounded function with an absolutely continuous function exists over any element in a field of subsets of a set U.

The purpose of this paper is to generalize a theorem that characterizes absolute continuity of bounded finitely additive set functions in the form of an integral approximation. We show that his integral exists if the condition of absolute continuity is removed.

The use of characteristic functions to represent well-known sets in graph theory such as dominating, irredundant, independent, covering and packing sets - leads naturally to fractional versions of these sets and corresponding fractional parameters. Let S be a dominating set of a graph G and f : V(G)~{0,1} the characteristic function of that set. By first translating the restrictions which define a dominating set from a set-based to a function-based form, and then allowing the function f to map the vertex set to the unit closed interval, we obtain the fractional generalisation of the dominating set S. In chapter 1, known domination-related parameters and their fractional generalisations are introduced, relations between them are investigated, and Gallai type results are derived. Particular attention is given to graphs with symmetry and to products of graphs. If instead of replacing the function f : V(G)~{0,1} with a function which maps the vertex set to the unit closed interval we introduce a function f' which maps the vertex set to {0, 1, ... ,k} (where k is some fixed, non-negative integer) and a corresponding change in the restrictions on the dominating set, we obtain a k-dominating function. In chapter 2 corresponding k-parameters are considered and are related to the classical and fractional parameters. The calculations of some well known fractional parameters are expressed as optimization problems involving the k- parameters. An e = 1 function is a function f : V(G)~[0,1] which obeys the restrictions that (i) every non-isolated vertex u is adjacent to some vertex v such that f(u)+f(v) = 1, and every isolated vertex w has f(w) = 1. In chapter 3 a theory of e = 1 functions and parameters is developed. Relationships are traced between e = 1 parameters and those previously introduced, some Gallai type results are derived for the e = 1 parameters, and e = 1 parameters are determined for several classes of graphs. The e = 1 theory is applied to derive new results about classical and fractional domination parameters. / Thesis (M.Sc.)-University of Natal, 1995.