 Projection-valued measure

A projection-valued measure $E$ assigns an orthogonal projection $E(B)$ to each measurable set, with $E(\varnothing )=0$, $E(X)=I$, $E(B\cap C)=E(B)E(C)$, and countable additivity in the strong operator topology on pairwise disjoint sets.

A projection-valued measure (PVM) is a fundamental concept in the fields of functional analysis and quantum mechanics, particularly in the mathematical formulation of quantum theory. It is a specific type of measure that assigns a projection operator to each measurable set in a given σ-algebra.