This eigenket equation says that if a measurement of the observable A ^ ^ is made while the system of interest is in the state | ψ ψ a ⟩ ⟩ , then the observed value of that particular measurement must return the eigenvalue a with certainty. However, if the system of interest is in the general state | ϕ ϕ ⟩ ⟩ ∈ ∈ H (and | ϕ ϕ ⟩ ⟩ and | ψ ψ a ⟩ ⟩ are unit vectors, and the eigenspace of a is one-dimensional), then the eigenvalue a is returned with probability | ⟨ ⟨ ψ ψ a | ϕ ϕ ⟩ ⟩ | 2 , by the Born rule.