This study of abstract measure covers classes of sets, measures and outer measures, extension of measures, and planer measure.

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.

Doctor of Philosophy / Department of Mathematics / Chris Pinner and Todd Cochrane

BACKGROUND:Health literacy (HL) is an important public health issue. Current measures have drawbacks in length and/or acceptability. The US-developed Newest Vital Sign (NVS) health literacy instrument measures both reading comprehension and numeracy skills using a nutrition label, takes 3 minutes to administer, and has proven to be acceptable to research subjects. This study aimed to amend and validate it for the UK population.METHODS:We used a three-stage process / (1) a Delphi study with academic and clinical experts to amend the NVS label to reflect UK nutrition labeling (2) community-based cognitive testing to assess and improve ease of understanding and acceptability of the test (3) validation of the NVS-UK against an accepted standard test of health literacy, the Test of Functional Health Literacy in Adults (TOFHLA) (Pearson's r and the area under the Receiver Operating Characteristic (ROC) curve) and participant educational level. A sample size calculation indicated that 250 participants would be required. Inclusion criteria were age 18-75 years and ability to converse in English. We excluded people working in the health field and those with impaired vision or inability to undertake the interview due to cognitive impairment or inability to converse in English.RESULTS:In the Delphi study, 28 experts reached consensus (3 cycles). Cognitive testing (80 participants) yielded an instrument that needed no further refinement. Validation testing (337 participants) showed high internal consistency (Cronbach's Alpha = 0.74). Validation against the TOFHLA demonstrated a Pearson's r of 0.49 and an area under the ROC curve of 0.81.CONCLUSIONS:The NVS-UK is a valid measure of HL. Its acceptability and ease of application makes it an ideal tool for use in the UK. It has potential uses in public health research including epidemiological surveys and randomized controlled trials, and in enabling practitioners to tailor care to patient need.

Research has shown that media exposure to violence is positively correlated with anxiety and posttraumatic stress symptoms in children (Becker-Blease et al. 2008), and parents can influence children’s responses to media (Otto et al. 2007). Few studies have examined specific parenting behaviors related to their children’s response to violent news media exposure; which is further limited by the lack of available measures with adequate psychometric support (Comer & Kendall, 2007). The current study addresses this gap by developing a measure of specific ways that caregivers may influence their children’s exposure and reaction to violent news. Item content was generated based on a literature review and focus group interview with six parents. Using a sample of 702 participants recruited through Amazon Mechanical Turk, exploratory factor analysis was conducted on the initial 74-item measure of Caregiver Responses to Youth Media Exposure (CRYME). With a total of 35 items, a three-factor solution emerged.

Large deviation theory has experienced much development and interest in the last two decades. A large deviation principle is the exponential decay of the probability of increasingly rare events and the computation of a rate or entropy function which measures the rate of decay. Within the probability literature there has been much use made of these rates in diverse applications. These large deviation principles have been discovered for independent and identically distributed random variables, as well as random vectors and these have been extended to some cases of weak dependence. In this thesis we prove large deviation principles for finite dimensional distributions of scaling limits of random measures. Functional approaches to large deviation theory using test functions as dual objects to random measures are also developed. These results are applied to some important classes of models, in particular Poisson point processes, Poisson center cluster processes and doubly stochastic point processes. / Graduation date: 1991

The definition of a groupoid is presented as well as examples of common structures from which a groupoid can be formed. Haar measure existence and uniqueness theorems for topological groups are used for the construction of Haar systems on groupoids. Some Haar systems are presented in addition to an example of a groupoid which admits no Haar system.

No description available.

This dissertation concerns two topics in analysis. The rst section is an exposition of the Henstock-Kurzweil integral leading to a necessary and su cient condition for the change of variables formula to hold, with implications for the change of variables formula for the Lebesgue integral. As a corollary, a necessary and suf- cient condition for the Fundamental Theorem of Calculus to hold for the HK integral is obtained. The second section concerns a challenge raised in a paper by O. Lazarev and E. H. Lieb, where they proved that, given f1….,fn ∈ L1 ([0,1] ; C), there exists a smooth function φ that takes values on the unit circle and annihilates span {f1...., fn}. We give an alternative proof of that fact that also shows the W1,1 norm of φ can be bounded by 5πn + 1. Answering a question raised by Lazarev and Lieb, we show that if p > 1 then there is no bound for the W1,p norm of any such multiplier in terms of the norms of f1...., fn. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2016. / FAU Electronic Theses and Dissertations Collection

Thesis (M.A.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / The thesis is divided into three parts. Part One, Measure and Measure Spaces, contains the introduction; and in later chapters this part approaches measure theory by the most natural route. It begins in a detailed fashion by developing the theory of set length through several levels of generalization, including the concepts of ordinary interval length, Jordan content, and Lebesgue measure. Then, without details, it discusses the extension of this theory of measure to more complex Euclidean sets and, finally, to abstract sets. Part Two, The Integral on Measure Spaces, introduces the theory of Lebesgue integration. The integrals treated gere are the measure dependent Lebesgue and general Lebesgue integrals. Chapter 6 in this part establishes a link between the Riemann integral and Jordan content, described in Chapter 3 of Part One, and between the Lebesgue integral and Lebesgue measure, described in Chapter 4 of Part One. Chapter 7 establishes an analogous link between the general Lebesgue integral and general measure spaces introduced in Chapter 5 of Part One. Chapter 8 , Measure on a Semialgebra, stands as a bridge between the historic and abstract versions of the Lebesgue Theory; for it generalizes the procedure used to construct the geometric integral, and shows that this procedure will produce general integrals as well. Part Three, A More General Integral , considers the theory of the Daniell integral. This integral is defined in more general terms than the measure-dependent Lebesgue integral of Part Two. Nevertheless, a harmonious resolution is reached in the theorems that show how the lines supporting the two theories can be drawn together / 2031-01-01