### Abstract:

It is well known that a commutative von Neumann algebra can be represented as a space of essentially bounded functions over a localizable measure space. In non-commutative integration theory, a von Neumann algebra takes over the role of the space of essentially bounded measurable functions. If the von Neumann algebra is semifinite, then there exists a faithful semifinite normal trace on it. Equipped with such a trace, a topology can be defined on the algebra, which in the commutative case is the familiar topology of convergence in measure. The completion of the algebra with respect to this topology yields an algebra of unbounded operators, the algebra of so-called measurable operators. In the first part of this thesis, the relationship between the nature of the lattice of projections of the Neumann algebra and the properties of this topology, in particular its local convexity, is investigated. In the duality theory for commutative Banach function spaces, one distinguishes between normal functionals and singular functionals. The study of the former leads to Koethe duality theory. A non-commutative Koethe duality theory already exists and a second aim of this thesis is to initiate a theory for singular functionals in the non-commutative setting. Finally, duality for the space of measurable operators equipped with the measure topology is investigated. Its Koethe dual is first characterised, and then singular functionals on this space are investigated. In cdertain cases a full characterization of the continuous dual is given.