probability that the observable takes on various values. A common choice for X is the real line, but it may also be R3 (for position or momentum in three dimensions ), a discrete set (for angular momentum, energy of a bound state, etc.), the 2-point set "true" and "false" for the truth-value of an arbitrary proposition about Φ. Let E be a measurable subset of the measurable space X and Φ a normalized vector-state in H, so that its Hilbert norm is unitary, ||Φ|| = 1. 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 (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. Second, for each fixed normalized vector state , the association is a probability measure on X making the values of the observable into a random variable. A measurement that can be performed by a projection-valued measure is called a projective measurement. If X is the real number line, there exists, associated to , a Hermitian operator A defined on H by which takes the more readable form if the support of is a discrete subset of R. The above operator A is called the observable associated with the spectral measure. Any operator so obtained is called an observable, in quantum mechanics. Generalizations The idea of a projection-valued measure is generalized by the positive operator-valued measure (POVM), where the need for the orthogonality implied by projection operators is replaced by the idea of a set of operators that are a non-orthogonal partition of unity. This generalization is motivated by applications to quantum information theory. See also Spectral theorem Spectral theory of compact operators Spectral theory of normal C*-algebras References Mackey, G. W., The Theory of Unitary Group Representations, The University of Chicago Press, 1976 M. Reed and B. Simon, Methods of Mathematical Physics, vols I–IV, Academic Press 1972. G. Teschl, Mathematical Methods in Quantum Mechanics with Applications to Schrödinger Operators, https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/, American Mathematical Society, 2009. Varadarajan, V. S., Geometry of Quantum Theory'' V2, Springer Verlag, 1970. Linear algebra Measures (measure theory) Spectral