In functional analysis and quantum measurement theory, a positive-operator valued measure (POVM) is a measure whose values are non-negative self-adjoint operators on a Hilbert space, and whose integral is the identity operator. It is the most general formulation of measurements in quantum physics. Since projective measurements on a large system—i.e., measurements that are performed mathematically by a projection-valued measure (PVM)—will act on a subsystem in ways that cannot be described by a PVM on the subsystem alone, the POVM formalism becomes necessary. POVMs are used in the field of quantum...
The following proposition comes from Lang's Algebraic Number Theory on page $12$:
Let $A$ be a ring integrally closed in its field of fractions $K$. Let $L$ be a finite Galois extension of $K$ with $G := \operatorname{Gal}(L/K)$. Let $\mathfrak{p}$ be a maximal ideal of $A$ and let $\mathfrak...
