Abstract model for measure and category

Abstract

This thesis is an attempt to establish an abstract model for Lebesgue measure and Baire category. In the introduction, we list several similarities between Lebesgue measurable sets and sets having the property of Baire. Then we abstract these similarities and use them as axioms. In Chapter I, we introduce a generalized model and prove some results that are well-known both in measure and category. In Chapter II, we define kernels and covers. After proving their existence for any set, we proceed to find some interesting results. It is very natural to consider the quotient algebra if we have an algebra containing a proper ideal. Hence Chapter III inevitably comes into the scene. In Chapter IV we introduce analytic sets through A-operations. This approach enables us to prove that every analytic set belongs to our model. In Chapter V we consider the local properties of sets and prove some interesting results. Chapter VI is taken from the work of J. C. Morgan II. We include his work here for the completeness of the thesis. Also, as we will see, it gives us a new insight into "negligible sets". We conclude the thesis by setting up a list of questions which, we think, are rather challenging.