of an observable \(A\), where each eigenvalue \(a_j\) of \(A\) occurs but once in the sum, \(j\neq k\) implies \(a_j\neq a_k\), associates with \(A\) a unique decomposition of the identity of the form \eqref{eqn5}. Thus \(A\) takes on or has the value \(a_j\) if and only if \(P_j\) is true, and the probability that \(A\) has the value \(a_j\) is given by the probability \(p_j\) assigned to \(P_j\). Hence measuring \(A\) is equivalent to measuring the corresponding decomposition in the sense of determining which of the mutually-exclusive properties \(P_j\) is true. (The subject of quantum measurements is taken up below in section 7.) The property that the value of \(A\) lies between \(a\) and \(a'\) corresponds to the sum of the projectors \(P_j\) for which the corresponding \(a_j\) falls in this interval, which is a projector belong to the corresponding event algebra, and its probability is the sum of the corresponding \(p_j\) values.