Suppose we are dealing with non relativistic quantum mechanics of point particles, that is we are in the realm of 'classic' quantum mechanics (no quantum fields ect.).
In the Heisenberg picture, observables are described by self adjoint operators, that act on a Hilbert space. Given a state and an observable, a measurement is modeled in such a way, that all eigenvalues of that operator are the possible outcomes.
An actual measurement then has one of the eigenvalues as its result and after the measurement the measured state vector is projected (collapsed if you like) onto the appropriate eigenvector of that eigenvalue.
On the other hand, the phase space description of quantum mechanics describes observables as functions on the phase space, but not with the ordinary 'dot' product (as in classical mechanics on the phase space) but with a $\star$-product. For example the Moyal product.
Now both descriptions are exactly equivalent via the Weil-Wigner transformation. That is there is a priori no reason to favor one over the other.
My question is: How is measurement and measurement outcome modeled in the $\star$-product phase space description of QM?