In my opinion, compactness is the most important concept in mathematics. We 'll track it from the one-dimensional real line in calculus to infinite dimensional spaces of functions and surfaces and see what it can do.

This paper considers the concept of inner and outer content, which was introduced by Camille Jordan and Giuseppe Peano near the end of the nineteenth century.

This dissertation is divided into two parts. The first part addresses the following problem: Suppose 𝑣 is a finitely additive probability measure defined on the power set 𝒜 of the integer Z so that each singleton set gets measure zero. Let X be a product space Π/β∈B * Zᵦ where each Zₐ is a copy of the integers. Let 𝒜ᴮ be the algebra of subsets of X generated by the subproducts Π/β∈B * Cᵦ where for all but finitely many β, Cᵦ = Zᵦ. Let 𝑣_B denote the product measure on 𝒜ᴮ which has each factor measure a copy of 𝑣. A subset E of X is said to be 𝑣_B -measurable iff [sic] there is only one finitely additive probability on the algebra generated by 𝒜ᴮ ∪ [E] which extends 𝑣_B. The set E ⊆ X is said to be universally product measurable (u.p.m.) iff [sic] for each finitely additive probability measure μ on 𝒜 which gives each singleton measure zero,E is μ_B -measurable. Two theorems are proved along with generalizations. The second part of this dissertation gives a proof of the following theorem and some generalizations: There are 2ᶜ nonisomorphic subalgebras of the power set algebra of the integers (where c = power of the continuum).

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The current trend among theorists in counseling and psychotherapy is toward the development of metatheoretical models that can be used to organize, systematically and comprehensively, existing theories and techniques within the discipline. Some models also provide behavior adaptation guidelines for practitioners who wish to adapt their behavior to client behavior patterns. Hutchins created the metatheoretical TFA System to accomplish the above goals. He also created the Hutchins Behavior Inventory (HBI) to complement the TFA System. The HBI purportedly measures the thinking, feeling, and acting dimensions of behavior upon which the TFA System is built; it thus enables a practitioner to assess the unique, situationally specific, TFA behavior pattern of a client. At the time of this study, the only evidence concerning the measurement properties of the HBI was for a form that produces ipsative scores (the HBI-I) . But ipsative scores possess inherent psychometric properties that cause problems when they are subjected to certain types of statistical analyses. Thus, in this study, a normative form of the HBI (the HBI-N) was designed. The measurement properties of the HBI-I and HBI-N were then investigated and compared. Reliability was investigated using test-retest and internal consistency procedures. Construct-related validity was investigated using four procedures: internal consistency analysis of HBI-N scores; factor analysis of the items comprising the scales of the HBI-N; an analysis of a multitrait-multimethod validity matrix containing scores from the HBI-I, HBI-N, Strong Campbell Interest Inventory (SCII), and Myers-Briggs Type Indicator (MBTI); and a factorial validity analysis of scores from the HBI-N, SCII, and MBTI. Results indicated that the HBI-I possesses a high degree of reliability. Prior evidence of content-related validity suggested that the three constructs measured by the HBI are the thinking, feeling, and acting dimensions of behavior hypothesized by Hutchins. Some of the construct-related validity results obtained in this study supported this conclusion, while the main body of results supported the more limited conclusion that the HBI scales measure different, yet to be more clearly identified, constructs. Based on the evidence in this study, the HBI-I seems appropriate for research and clinical use. / Ph. D.

In this paper we will present two different approaches to the development of product measures. In the second chapter we follow the lead of H. L. Royden in his book Real Analysis and develop product measure in the context of outer measure. The approach in the third and fourth chapters will be the one taken by N. Dunford and J. Schwartz in their book Linear Operators Part I. Specifically, in the fourth chapter, product measures arise almost entirely as a consequence of integration theory. Both developments culminate with proofs of well known theorems due to Fubini and Tonelli.

In this work, we concern ourselves with particular topics in Polish space theory. We first consider the space A(U) of complex-analytic functions on an open set U endowed with the usual topology of uniform convergence on compact subsets. With the operations of point-wise addition and point-wise multiplication, A(U) is a Polish ring. Inspired by L. Bers' algebraic characterization of the relation of conformality, we show that the topology on A(U) is the only Polish topology for which A(U) is a Polish ring for a large class of U. This class of U includes simply connected regions, simply connected regions excluding a relatively discrete set of points, and other domains of usual interest. One thing that we deduce from this is that, even though C has many different Polish field topologies, as long as it sits inside another Polish ring with enough complex-analytic functions, it must have its usual topology. In a different direction, we show that the bounded complex-analytic functions on the unit disk admits no Polish topology for which it is a Polish ring. We also study the Lie ring structure on A(U) which turns out to be a Polish Lie ring with the usual topology. In this case, we restrict our attention to those domains U that are connected. We extend a result of I. Amemiya to see that the Lie ring structure is determined by the conformal structure of U. In a similar vein to our ring considerations, we see that, again for certain domains U of usual interest, the Lie ring A(U) has a unique Polish topology for which it is a Polish Lie ring. Again, the Lie ring A(U) imposes topological restrictions on C. That is, C must have its usual topology when sitting inside any Polish Lie ring isomorphic to A(U). In the last chapter, we introduce a new ideal of subsets of Polish spaces consisting of what we call residually null sets. From this ideal, we introduce an algebra consisting of what we call R-sets which is consistently a strict extension of the algebra of Baire property sets. We show that the algebra of R-sets is closed under the Alexandrov-Suslin operation and generalize Pettis' Theorem. From this, we provide new automatic continuity results and give a generalization of a result of D. Montgomery which shows that minimal assumptions on the continuity of group operations of an abstract group G with a Polish topology imply that G is actually a Polish group. We also see that many results pertaining to the algebra of Baire property sets generalize to the context of R-sets.