Repeated measurements in quantum theory

Limiting formulas are deduced for the state of a quantum system which is subjected to a rapid sequence of many measurements of the same two-valued observable. In the case of a semibounded Hamiltonian, the dynamics usually gets reduced to the subspace spanned by the eigenstates of the observable. In particular, the values of the observable "freeze." Exceptions to freezing occur for states which cannot be approximated by eigenstates of finite energy.