| This thesis contains a few applications of set-theoretic methods to certain problems in real analysis. In the first two sections of Chapter 1, we discuss some results related to a question of Fremlin about partitions of a set of reals into null sets. In Section 3, we answer a questions of Komjath in dimension one. Our proof uses some results of Gitik and Shelah in an essential way. In Chapter 2, we answer a couple of questions about finitely additive total extensions of Lebesgue measure. These problems arose from a question of Juhasz in set-theoretic topology. In Chpater 3, we give some "natural" examples of additive subgroups of reals of arbitrarily high finite Borel rank. The existence of such groups is an old and well known result. In Chapter 4, we construct a non principal ultrafilter from any free maximal ideal in the ring of bounded continuous functions on reals. |
