Compatibility of observables represented by positive operator-valued measures

The proof of a result analogous to that in Koelman and de Muynck Phys. Lett. A 98, 1 (1983) is given for the case of unbounded observables. If two, not necessarily bounded, observables are represented by a positive operator-valued measure, then the measurement of any of them is undisturbed if and only if they commute. The Naimark theorem on dilations of spectral functions is exploited. A stronger version of Wigner’s theorem is given.