The ideal of definition of a
faithful semifinite normal weight on a countably decomposable von Neumann algebra
is the set generated by all positive elements of finite weight. The set is a hereditary
left ideal and therefore contains projections. In this paper the family of weights
whose ideals of definition form projection lattices is completely characterized. These
weights are the ones that are comparable to a combination of traces and normal
functionals. A central spectral resolution is introduced and used to analyze the
Radon-Nikodym derivatives of a weight with regard to a trace. Also introduced are
two parameters that measure whether the ideal of definition contains two
projections of least upper bound 1 and how close the weight is to being a trace
respectively.
