  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 quantum theory to define the probability of equality relation
for a pair of arbitrary observables. Applications of this new interpretation to
measurement theory are discussed briefly.
