This thesis examines measure functions. A measure function has as its domain of definition a class of sets. It also must satisfy a certain additive condition. To state a concise definition of a measure function, it is convenient to define set function and completely additive set function.

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.

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Motivated by the work of Hognas and Mukherjea on semigroups,we study semihypergroups, which are structures closer to semigroups than hypergroups in the sense that they do not require an identity or an involution. Dunkl[Du73] calls them hypergroups (without involution), and Jewett[Je75] calls them semiconvos. A semihypergroup does not assume any algebraic operation on itself. To generalize results from semigroups to semihypergroups, we first put together the fundamental algebraic concept a semihypergroup inherits from its measure algebra. Among other things, we define the Rees convolution product, and prove that if X; Y are non-empty sets and H is a hypergroup, then with the Rees convolution product, X x H x Y is a completely simple semihypergroup which has all its idempotent elements in its center. We also point out striking differences between semigroups and semihypergroups. For instance, we construct an example of a commutative simple semihypergroup, which is not completely simple. In a commutative semihypergroup S, we solve the Choquet equation μ * v = v, under certain mild conditions.We also give the most general result for the non-commutative case.We give an example of an idempotent measure on a commutative semihypergroup whose support does not contain an idempotent element and so could not be completely simple. This is in contrast with the context of semigroups, where idempotent measures have completely simple supports. The results of Hognas and Mukherjea [HM95] on the weak convergence of the sequence of averages of convolution powers of probability measures is generalized to semihypergroups. We use these results to give an alternative method of solving the Choquet equation on hypergroups (which was initially solved in [BH95] with many steps). We show that If S is a compact semihypergroup and μ is a probability measure with S = [ U∞n=1 Supp(μ)n], then for any open set G ⊃ K where K is the kernel of S limn-→∞μn(G) = 1. Finally, we extend to hypergroups basic techniques on multipliers set forth for groups in [HR70], namely propositions 5.2.1 and 5.2.2 , we give a proof of an extended version of Wendel's theorem for locally compact commutative hypergroups and show that this version also holds for compact non-commutative hypergroups. For a compact commutative hypergroup H, we establish relationships between semigroup S = S = {T(ξ) : ξ > 0} of operators on Lp(H), 1 ≤ p < 1 < ∞, which commutes with translations, and semigroup M = {Eξ : ξ > 0} of Lp(H) multipliers. These results generalize those of [HP57] for the circle groups and [B074] for compact abelian groups.

Neste trabalho fizemos uma revisão geral e encontramos resultados novos sobre a simetria unitária simplética. Obtivemos uma fórmula simples para a exponencial da álgebra de Lie simplética complexa em quatro dimensões, sp(4, C). A partir da decomposição de Gauss do referido grupo, impusemos a unitariedade para obtermos expressões analíticas para esta decomposição. Ao impormos a condição unitária ao grupo simplético, formamos o grupo unitário simplético e obtivemos as regras de multiplicação deste grupo, as quais estão implementadas simbólicamente tendo em mente aplicações futuras. Como uma consequência encontramos uma representação da álgebra de Lie em termos de operadores diferenciais. Uma segunda e mais importante conseqüência foi a obtenção da métrica de Haar deste grupo, a qual é fundamental no estudo dos estados coerentes. Um rápido estudo da quebra de simetria entre a cadeia canônica e a cadeia de termos Majorana é apresentada no apêndice tendo em vista futuras aplicações ao estudo algébrico do código genético. Os estados coerentes do grupo Usp(4) foram calculados para uma representação arbitrária e a supercompleteza foi demonstrada devido a métrica de Haar, isto completa o programa iniciado por Novaes em sua tese de PhD. Os valores médios dos geradores da álgebra de Lie foram obtidos tendo em mente a aplicação a um hamiltoniano algébrico. Por fim, obtivemos a forma simplética numa representação arbitrária, preparando o campo para aplicações aos sistemas dinâmicos. / In this work we take a general revision and take new results on the unitary symplectic symmetry. We have obtained a simple form for the exponential of the complex symplectic Lie algebra on four dimensions, sp(4, C). With the Gauss decomposition for this group, we impose the unitarity to obtain analytical expressions for that Gauss decomposition. Imposing the analytical expressions to the Gauss decomposition for the complex symplectic algebra, we have been obtained explicit multiplication formulas for the unitarian group and iinplemented symbolically have in mind further application. As a consequence a representation of the Lie algebra in terms of differential operators have been obtained. The Haar measure that plays a fundamental role in the study of coherent states is calculated in an arbitrary representation. An early study envolving the symmetry breaking of canonical Sp(4) tree by Majorana operators is presented in the appendix in the spirit of algebraic approach to genetic code. The coherent states of USp (4) have been calculated for an arbitrary representation and the overcompletness is demonstred thanks to the Haar measure, the program initiate by Novaes in his PhD thesis is now fully completed. The mean values of the Lie algebra generators in a coherent state base are calculated having in mind application to algebraic hamiltonian. Finally we obtained the symplectic form in a arbitrary representation have also been calculate preparing the field for applications to dynamical systems.

Neste trabalho fizemos uma revisão geral e encontramos resultados novos sobre a simetria unitária simplética. Obtivemos uma fórmula simples para a exponencial da álgebra de Lie simplética complexa em quatro dimensões, sp(4, C). A partir da decomposição de Gauss do referido grupo, impusemos a unitariedade para obtermos expressões analíticas para esta decomposição. Ao impormos a condição unitária ao grupo simplético, formamos o grupo unitário simplético e obtivemos as regras de multiplicação deste grupo, as quais estão implementadas simbólicamente tendo em mente aplicações futuras. Como uma consequência encontramos uma representação da álgebra de Lie em termos de operadores diferenciais. Uma segunda e mais importante conseqüência foi a obtenção da métrica de Haar deste grupo, a qual é fundamental no estudo dos estados coerentes. Um rápido estudo da quebra de simetria entre a cadeia canônica e a cadeia de termos Majorana é apresentada no apêndice tendo em vista futuras aplicações ao estudo algébrico do código genético. Os estados coerentes do grupo Usp(4) foram calculados para uma representação arbitrária e a supercompleteza foi demonstrada devido a métrica de Haar, isto completa o programa iniciado por Novaes em sua tese de PhD. Os valores médios dos geradores da álgebra de Lie foram obtidos tendo em mente a aplicação a um hamiltoniano algébrico. Por fim, obtivemos a forma simplética numa representação arbitrária, preparando o campo para aplicações aos sistemas dinâmicos. / In this work we take a general revision and take new results on the unitary symplectic symmetry. We have obtained a simple form for the exponential of the complex symplectic Lie algebra on four dimensions, sp(4, C). With the Gauss decomposition for this group, we impose the unitarity to obtain analytical expressions for that Gauss decomposition. Imposing the analytical expressions to the Gauss decomposition for the complex symplectic algebra, we have been obtained explicit multiplication formulas for the unitarian group and iinplemented symbolically have in mind further application. As a consequence a representation of the Lie algebra in terms of differential operators have been obtained. The Haar measure that plays a fundamental role in the study of coherent states is calculated in an arbitrary representation. An early study envolving the symmetry breaking of canonical Sp(4) tree by Majorana operators is presented in the appendix in the spirit of algebraic approach to genetic code. The coherent states of USp (4) have been calculated for an arbitrary representation and the overcompletness is demonstred thanks to the Haar measure, the program initiate by Novaes in his PhD thesis is now fully completed. The mean values of the Lie algebra generators in a coherent state base are calculated having in mind application to algebraic hamiltonian. Finally we obtained the symplectic form in a arbitrary representation have also been calculate preparing the field for applications to dynamical systems.

Magistro darbe nagrinėjamas Oilerio sandaugų reikšmių pasiskirstymas analizinių funkcijų erdvėje. Taip pat gaunamas išreikštinis ribinio mato pavidalas. / In the Master work, we investigate the value-distribution of Euler products in the space of analytic functions. Also, we give an explicit form of the limit measure.