In 1998, it was proved by Ben-David and Litman that a concept space has a sample compression scheme of size $d$ if and only if every finite subspace has a sample compression scheme of size $d$. In the compactness theorem, measurability of the hypotheses of the created sample compression scheme is not guaranteed; at the same time measurability of the hypotheses is a necessary condition for learnability. In this thesis we discuss when a sample compression scheme, created from compression schemes on finite subspaces via the compactness theorem, have measurable hypotheses. We show that if $X$ is a standard Borel space with a $d$-maximum and universally separable concept class $\m{C}$, then $(X,\CC)$ has a sample compression scheme of size $d$ with universally Borel measurable hypotheses. Additionally we introduce a new variant of compression scheme called a copy sample compression scheme.

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.

The aim of this study is to develop new knowledge about the challenges and possibilities of Bildung in the Swedish compulsory school. Based on a qualitative analysis of five elite interviews with five theorists of Bildung, this study is inspired by curriculum theory and, in particular, the key analytical concepts of curriculum models, frame factors and room for freedom The results make visible both challenges and possibilities for Bildung in the Swedish compulsory school.  First, through a thematic analysis, five themes were identified: i) Bildung as an open process; ii) Bildung in relation to coherence and overview; iii) Bildung and meaning making; iv) Bildung in education; and v) Bildung as resistance. This thematic analysis has shown that tensions exist between Bildung and the measurable results that are the focus in the Swedish school. Furthermore, a one-sided alignment to measurability tends to neglect several dimensions of Bildung.  Second, the outcomes from the thematic analysis were considered in light of the curriculum theory framework in the form of an additional analysis based on three theoretical curriculum models. Using the theoretical concepts frame factors and room for freedom, the analysis demonstrates that various curriculum models, in different ways, both restrict Bildung and make it possible. Despite this, the results illustrate how Bildung, to a greater or lesser extent, can be operationalized through different curriculum models.  The third analysis reveals a shift in the understanding of Bildung in the Swedish school, from something differential to something for every pupil. The final results are visualized as three “windows” of Bildung, each of which presents how Bildung can be a natural component of the Swedish compulsory school: i) a democratized and expanded view of Bildung; ii) a fluid dialogue between the subjective and objective dimensions of Bildung; and iii) Bildung as resistance.  This study contributes to a current and forward-looking discussion about the role of Bildung, with a special focus on the Swedish compulsory school; however, it is also of relevance outside the Swedish context.

In 1998, it was proved by Ben-David and Litman that a concept space has a sample compression scheme of size $d$ if and only if every finite subspace has a sample compression scheme of size $d$. In the compactness theorem, measurability of the hypotheses of the created sample compression scheme is not guaranteed; at the same time measurability of the hypotheses is a necessary condition for learnability. In this thesis we discuss when a sample compression scheme, created from compression schemes on finite subspaces via the compactness theorem, have measurable hypotheses. We show that if $X$ is a standard Borel space with a $d$-maximum and universally separable concept class $\m{C}$, then $(X,\CC)$ has a sample compression scheme of size $d$ with universally Borel measurable hypotheses. Additionally we introduce a new variant of compression scheme called a copy sample compression scheme.

In 1998, it was proved by Ben-David and Litman that a concept space has a sample compression scheme of size $d$ if and only if every finite subspace has a sample compression scheme of size $d$. In the compactness theorem, measurability of the hypotheses of the created sample compression scheme is not guaranteed; at the same time measurability of the hypotheses is a necessary condition for learnability. In this thesis we discuss when a sample compression scheme, created from compression schemes on finite subspaces via the compactness theorem, have measurable hypotheses. We show that if $X$ is a standard Borel space with a $d$-maximum and universally separable concept class $\m{C}$, then $(X,\CC)$ has a sample compression scheme of size $d$ with universally Borel measurable hypotheses. Additionally we introduce a new variant of compression scheme called a copy sample compression scheme.

ii Abstract The topic of his thesis, "Social Sustainability", arising from the social sciences, specifically sociology, has naturally its core focus on society and its functioning. To be more exact, the functioning of society in context of integration, which is an ever more growing topic in the globalised world we live in. In relation to this thesis, an integrational project/workshop has been carried out at a folk high school, Brandbjerg, in Denmark, with the purpose to practically test, the methodological measurability of change in subjectivity. Worth mentioning, that the hosting community, in terms of immigration, was in focus and its perception on variation of habitus. Hereto, Q-Methodology from William Stephenson functioned as the foundation for the development of the workshop, as well as it aided the analysis of the therefrom taken data, to determine a change in the participants subjectivity. The aim of this practical implementation, was to put the findings into perspective of the theoretical framework of sustainability, specifically social sustainability. The findings served in combination with face to face interaction as part of social integration, to underline the need for sustainable integration and an alteration of social sustainability upon further research. A tendency was determined as being the...

I detta kandidatarbete tar vi upp om kopplingen mellan digitaliseringen och mätbarhet samt dess konsekvenser på den individuella människan, vilket vi sedan anpassat och implementerat i ett digitalt spel för att öppna upp för diskussion angående ämnet. Critical design är ett huvudfokus för vårt kandidatarbete för att kunna lyckas skapa diskussion och uppmana spelaren till att tänka på sina handlingar. I texten beskriver vi även hur spelet utvecklats, från idé och koncept, genom produktionsfasen fram till hur slutprodukten blev till. Vidare har vi även intervjuat för att se hur spelet samt dess kopplingar med digitalisering och mätbarhet uppfattades. Till detta ändamål kombinerades observationer, kvalitativa intervjuer, kvantitativa data och meningskoncentrering. Slutresultatet var inte riktigt som vi förväntat oss, digitaliseringen var inte så starkt representerad som vi hade hoppats på men mätbarhet och stress var det dock. Som avslutning följs det en diskussion kring vad som kunde ha förändrats samt vad framtida projekt skulle kunna ta lärdomar av. / In this bachelor thesis we address the connection between digitalization and measurability and its consequences on the individual, which we have adapted and implemented into a digital game to open up for a discussion about the subject. Critical Design is a main focus for our bachelor thesis in order to succeed with creating a discussion and challenge the players to think about their actions. In the text we also describe how the game was developed, from idea and concept, through the production phase up until the final product. Furthermore we’ve also performed interviews to understand how the game and its connections with digitalization and measurability was perceived. For this purpose, observations, qualitative interviews, quantitative data and sentence concentrating were combined. The end result wasn’t as we had imagined, digitizing wasn’t as strongly represented as we had hoped for but measurability and stress were. Lastly we’ll have a discussion about what could’ve changed and also what future projects could learn from this project.

Random measurable sets and particle processes Adam Jurčo Abstract In this thesis we deal with particle processes on more general spaces. First we in- troduce the space of Lebesgue measurable sets represented by indicator functions with topology given by L1 loc convergence. We the explore the topological properties of this space and its subspaces of sets of finite and locally finite perimeter. As these spaces do not satisfy the usual topological assumptions needed for construction of point processes we use another approach based on measure-theoretic assumptions. This will allow us to define point processes given by finite dimensional distributions on measurable subsets of the space of Lebesgue-measurable sets. Then we will derive a formula for a volume fraction of a Boolean process defined in this more general setting. Further we introduce a Boolean process with particles of finite perimeter and derive a formula for its specific perimeter. 1

In this thesis we shall examine the role of measurability in the theory of Lebesgue Integration. This shall be done in the context of the real line where we define the notion of an integral of a bouuded real-valued function over a set of bounded outer measure without a prior assumption of measurability concerning the function and the domain of integration

Statistical data depth is a nonparametric tool applicable to multivariate datasets in an attempt to generalize quantiles to complex data such as random vectors, random functions, or distributions on manifolds and graphs. The main idea is, for a general multivariate space M, to assign to a point x ∈ M and a probability distribution P on M a number D(x; P) ∈ [0, 1] characterizing how "centrally located" x is with respect to P. A point maximizing D(·; P) is then a generalization of the median to M-valued data, and the locus of points whose depth value is greater than a certain threshold constitutes the inner depth-quantile region corresponding to P. In this work, we focus on data depth designed for infinite-dimensional spaces M and functional data. Initially, a review of depth functionals available in the literature is given. The emphasis of the exposition is put on the unification of these diverse concepts from the theoretical point of view. It is shown that most of the established depths fall into the general framework of projection-driven functionals of either integrated, or infimal type. Based on the proposed methodology, characteristics and theoretical properties of all these depths can be evaluated simultaneously. The first part of the work is devoted to the investigation of these theoretical properties,...