This is a phantastic example for the strange inability of mathematicians and physicists to communicate with each other. I must admit, I lost track what's the problem with "degeneracy" here. There is no such problem from a physicist's point of view, because it's all well defined in terms of observables (discribed by self-adjoint operators of Hilbert space) and states (described by a self-adjoint trace-class operator with trace 1, the statistical operator):
Any complete set of independent compatible observables ##A_j## (##j \in \{1,2,\ldots,n\}##) defines via the common (generalized) eigenvectors of the corresponding self-adjoint operators ##\hat{A}_j## (that are pairwise commuting by definition), denoted by ##|a_1,\ldots,a_n \rangle##. Here the the components of the tupel ##(a_1,\ldots,a_n) \in \mathbb{R}^n## can run over discrete and/or continuous domains.
If the state of the system is represented by the statistical operator ##\hat{\rho}## the probability/probability density to simultaneously find the values ##(a_1,\ldots,a_n)## measuring the observables ##A_j## is given by (Born's rule):
$$P_{\rho}(a_1,\ldots,a_n)=\langle a_1,\ldots,a_n|\hat{\rho}|a_1,\ldots,a_n \rangle.$$
Everything else follows from the standard rules of probability theory. If I measure only von observable, e.g., ##A_1##, I just have to sum/integrate over all other ##a_2,\ldots,a_n## over the corresponding spectrum, i.e.,
$$P_{\rho}(a_1)=\sum_{a_2,\ldots,a_n} P_{\rho}(a_1,\ldots,a_n).$$
This refers to measuring precisely the observable ##A_1##. Of course in this case the eigenspaces of ##\hat{A}_1## alone are degenerate, because you need the other independent compatible observables to completely define the orthonormalized basis vectors (up to phase factors for each basis vector of course, but these cancel always out in physical results that are all dictated by the above given probability measure according to Born's rule). QT as a mathematical theory is consistent, as "weird" as some physicists and mathematicians seem to think it might be ;-))!
If you measure something not precisely you can envoke more complicated descriptions in terms of POVMs. Strictly speaking that you must do in any case if you measure an observable in the continuous part of the spectrum, because then you necessarily must use an apparatus with finite resolution, i.e., you can make, e.g., a position measurement of a particle, only up to a certain resolution determined by a the technology available for the measurement apparatus (which can be as fine as you like but not absolutely exact ever).
Now you can of course make this into a precise and well-defined mathematical formalism, and this is nice too, but one should not forget this not that difficult to understand physics and "metrological" constraints, which are very clear to everybody who as ever done experiments (and be it as a theoretician only in the standard lab practice mandatory to attend for every physics student wherever physics is taught).