The notion of equality between two observables will play many important roles in
foundations of quantum theory. However, the standard probabilistic interpretation based on
the conventional Born formula does not give the probability of equality between two arbitrary
observables, since the Born formula gives the probability distribution only for a commuting
family of observables. In this paper, quantum set theory developed by Takeuti and the
present author is used to systematically extend the standard probabilistic interpretation of …

The notion of equality between two observables will play many important roles in
foundations of quantum theory. However, the standard probabilistic interpretation based on
the conventional Born formula does not give the probability of equality relation for a pair of
arbitrary observables, since the Born formula gives the probability distribution only for a
commuting family of observables. In this paper, quantum set theory developed by Takeuti
and the present author is used to systematically extend the probabilistic interpretation of …