Spaces of complex functions and vector measures in incomplete spaces

It is known that the space L1(μ) of complex functions which are integrable with respect to a vector measure μ taking values in a (not neessarily complete) locally convex space is not an ideal, in general. We discuss several natural properties which L1(μ) may or may not possess and consider various implications between these properties. For a particular class of properties, whether or not there exists a partiular space of the form L1(μ) having these properties, is shown to be equivalent to the existence of any space of complex functions on C having these same properties.