Studies On Some Classes Of Measures On Effect Algebras

Quantum theory is one of greatest achievements in the twentieth century. Quantumlogic theory, which has developed during the course of the mathematical axiomatizationof the quantum theory, has more than 80 years'history and rich contents. Since 1936Birkhoff and von Neumann proposed the concept of quantum logics, the lattices of or-thogonal projections in separable complex Hilbert spaces have been regarded as the mostimportant mathematical model of quantum logics. However, this kind of lattices just coulddescribe the sharp phenomena. In 1994, Foulis and Bennett introduced a mathematicalstructure, which is called effect algebra, to research a general structure of the mathemat-ical foundation of quantum mechanics. It is a significant progress for the mathematicalaxiomatization of the quantum theory. As we know, measure theory, as an importantbranch of mathematical sciences, is an important tools of modern analytical mathemat-ics. It is in close contact with many parts of mathematics such as set theory, probabilitytheory, real analysis, differential equations, and it is extensively used in others areas ofnon-mathematics. Therefore, studying measure theory on quantum logic has not onlytheoretical significance but also practical values.This thesis consists of four chapters. We brie?y present a survey of over 80 years'study of quantum logic theory and the study history as well as present situation in mea-sure theory related to this paper. We investigate mainly bounded variation measuresand strongly bounded measures on effect algebras, the convergence of one kind of non-additive measures and continuities of monotone non-additive measures. The main con-tents are as follows:1. We prove that the equivalence of bounded measures, strongly additive measuresand strongly bounded measures defined on effect algebras taking values in real numberfield, and study bounded variation measures and give some elementary properties withrespect to them. At the same time, we prove that bounded variation measure spacesand bounded chain variation measure spaces are both Banach spaces. Also we introduceand research two properties of strongly bounded measure sequences and the relationshipbetween them and uniformly convergence theorem. Besides, we point out that these prop-erties are weaker than uniformly strong boundedness of measure sequences by giving a counterexample.2. We research a kind of non-additive measures on SCP-effect algebras, i.e., s-outermeasures, and establish Drewnowski lemma for such class of non-additive measures. Fur-thermore, by the lemma, we obtain the Brooks-Jewett theorem and extend its applicablescope.3. We discuss some kinds of continuities of non-additive measures on lattice effectalgebras, i.e., order continuity, continuity, strong continuity, autocontinuities, monotoneautocontinuities from above and below, respectively. Moreover, we prove that monotoneorder continuous non-additive measures on SCP-effect algebras are strongly bounded andgive one equivalent condition. In the last of this chapter, we present some examples toshow the relationship between order continuity, autocontinuity from above (below) andmonotone autocontinuity from above (below).