Short answer: this is often a problem of measure theory (see Measure Theory, by Paul Halmos), which by working with the grand-canonical ensemble can be bypassed. I have discussed this issue in Appendix C of my following paper:

http://arxiv.org/abs/0711.0952

See in particular the discussions on pages 148-154. In essence, by introducing the microcanonical density of states (Eq. (C.33)), together with employing some ansatz (Eq. (C.25)), one can get rid of some summations over very complicated quantum degrees of freedom whose evaluation would not be feasible by working in other ensembles.