A more general formulation replaces the projection-valued measure with a positive-operator valued measure (POVM). To illustrate, take again the finite-dimensional case. Here we would replace the rank-1 projections | ψ ψ i ⟩ ⟩ ⟨ ⟨ ψ ψ i | by a finite set of positive operators F i F i ∗ ∗ whose sum is still the identity operator as before (the resolution of identity). Just as a set of possible outcomes { λ 1 ... λ n } is associated to a projection-valued measure, the same can be said for a POVM. Suppose the measurement outcome is λ i. Instead of collapsing to the (unnormalized) state | ψ ψ i ⟩ ⟩ ⟨ ⟨ ψ ψ i | ψ ψ ⟩ ⟩ after the measurement, the system now will be in the state F i | ψ ψ ⟩ ⟩ .