 The probability that the observable takes its value in the subset "E," given the system in state Φ, is

where the latter notation is preferred in physics.We can parse this in two ways.First, for each fixed "E", the projection π("E") is a self-adjoint operator on "H" whose 1-eigenspace is the states Φ for which the value of the observable always lies in "E", and whose 0-eigenspace is the states Φ for which the value of the observable never lies in "E".