  The joint Brown measure and joint Haagerup--Schultz projections for tuples of
commuting operators in a von Neumann algebra equipped with a faithful tracial
state are investigated, and several natural properties are proved for these. It
is shown that the support of the joint Brown measure is contained in the Taylor
joint spectrum of the tuple, and also in the ostensibly smaller left Harte
spectrum. A simultaneous upper triangularization result for finite commuting
tuples is proved and the joint Brown measure and joint Haagerup--Schultz
projections are shown to be have well under the Arens multivariate holomorphic
functional calculus of such a commuting tuple.
