MEASUREMENT, INTEGRATION AND AREA

Author: CAMPO ACOSTA RICARDO DEL.
Year: 2004.
University: ALMERÍA [www.ual.es].
Place of defense: FACULTAD DE CIENCIAS EXPERIMENTALES.
Place of preparation: FACULTAD DE CIENCIAS EXPERIMENTALES.

Summary: Our work fits into the environment Integration Finitamente Additive investigation. In particular we made contributions in three directions: After a first chapter with prelimínares necessary in order to continue reading from memory, first studying under what conditions the integration abstract Riemann supports a comprehensive representation through joint spectral (Cápítulo 2). In particular test, in this environment functional, still ensuring that any abstract Riemann Integral is casimedible and that conditions of weak continuity are sufficient to generalize to get the formula for calculating Topsoe given by the integral of a role through measures their spectral sets. After examining the relationship of the absolute continuity for funcíonales, in the context of íntegración own abstract and Riemann, with its eponymous property for measures finitamente ADDITIVE, giving results in both senses-coming comprehensive measures and actions induced integrals (Chapter 3). In addition we developed some novel techniques density sequential allowing us to get a theorem Radon - Nikodym approximate functional in this environment (Chapter 4). Lastly, we developed a theory regarding comprehensive higher integration so that gives us a general atmosphere from which to address the Theory of Integration Finitamente Additive in a global (Chapter 5). Specifically, is the new concept of bideterminación we have introduced which has enabled us to find a common framework from which to study, simultaneously, certain aspects of several theories of integration that until now had been treated separately, but showed a certain parallelism formal.

Author: AMARAL ABREU TERESA PAULA.
Year: 2006.
University: VIGO [www.uvigo.es].
Place of defense: E.T.S.E. INDUSTRIALES.
Place of preparation: E.T.S.E. TELECOMUNICACIONES.

Summary: The main objective of this thesis is to extend the schema of Lebesgue integration of functions defined in a space premedida and valued in a conoide quasi-uniforme and also contains interesting contributions to the general theory of spaces quasi-uniformes, theory structures algebraic quasi-uniformes and the theory of measures valued at these structures, it is necessary to develop, in advance, to achieve the aforementioned objective. In the thesis choosing a very natural way for the presentation of the theory of space quasi-uniformes. The rise in the broader situation of the local quasi-uniformidades and is not supposed known theory spaces uniform, the results of which are recovered as special cases where there is symmetry. Introducing the new spaces bilocal quasi-uniformes and clarifies the problem of the tiny structures uniform rate and the topology induced them sensitive issue that usually does not appear in the literature. The thesis contains a full presentation of the theory of the structures algebraic quasi-uniformes including the basics of conoides (monoides equipped with a real multiplication by non-negative) quasi-uniformes which constitute the natural range of functions measurable, action and integrals. A noteworthy contribution in this part is the solution to the problem of negative local quasi-uniformación of semigrupos topological. In the thesis A. Castejón (1995), also directed by E. Corbacho, spread the theory of Lebesgue integration features valued at conoides with a consistent metrics positively homogeneous. In this thesis T. Abreu extends this possibility in the case of conoides quasi-uniformes. It proves that the presence of ownership of uniqueness may develop comprehensive scheme Lebesgue even without assuming symmetry. The fact that the hyperspace of subassemblies compact convex a conoide quasi-uniforme, with the quasi-uniformidad of Hausdorff-Borubaki, is also a conoide quasi-uniforme, this thesis provides a wide field of applications.