Throughout this chapter we will assume (unless stated otherwise) that (Ω,F,μ) is a σ-finite measure space, and that the σ-field F is complete with respect to μ. This implies that if f:Ω→R*, g: Ω→R* are functions such that f is F-measurable and f = g a.e., then g is also F-measurable. Thus, if M is the class of functions f: Ω → R* which are F-measurable, we say that f1,f2 in M are equivalent if f1 = f2 a.e. This clearly defines an equivalence relation in M and we can form the space M of equivalence classes with respect to this relation. When we think of a function f of M as an element of M we are really thinking of f as a representative of the class of F-measurable functions which are equal to fa.e. As is usual we will use the same notation f for an element of M and M. We can think of M or M as an abstract space, and the definition of convergence if given in terms of a metric will then impose a topological structure on the space. We will consider several such notions of convergence of which some, but not all, can be expressed in terms of a metric in M. We will obtain the relationships between different notions of convergence, and in each case prove that the space is complete in the sense that for any Cauchy sequence there is a limit function to which the sequence converges.