Some observations and results concerning submeasures on Boolean algebras

We investigate submeasures on Boolean algebras in the context of Maharam's problem and its solution. We generalise results that were originally proved for measures, to cases where additivity is not present. We investigate Talagrand's construction of a pathological exhaustive submeasure, attempting to give a more explicit description of this submeasure and we also consider some of its forcing properties. We consider the forcing consisting of submeasures that have as their domain a �nite subalgebra of the countable atomless Boolean algebra. We �nd and investigate a linear association between the real vector space of all real-valued functionals on the countable atomless Boolean algebra, which includes the collection of all submeasures, and the space of all signed �nitely additive measures on this Boolean algebra