Smeared Wigner functions and quantum-mechanical histories

We calculate the probability for a quantum-mechanical history consisting of imprecise samplings of position at two moments of time. In the limit of small time separation, this leads to an imprecise sampling of position together with a time-of-flight sampling of momentum. We also calculate the probability for the history consisting of direct momentum and position samplings a short time apart. In each case, we find that the resulting probability distribution on phase space is a smeared version of the Wigner function, and is positive. We show that these smearings belong to a class of smearings which make the Wigner function positive. In the case of the time-of-flight momentum sampling, it is more general than previously considered smearings, such as that of Husimi.