Ideals of Projections According to σ-Algebras and Unbounded Measurements

Abstract

A theory of unbounded measures is constructed based on the quantum logics of orthogonal projections. As an analogue of the ring of sets, the projector ideal is proposed. Finite and maximal measures regarding the projector ideals are described. Analogues of a number of classical theorems of measure theory are found. A wide class of unbounded measures on projection ideals is characterized. A number of sufficient conditions are found to extend unbounded measures to an integral of the entire algebra. The problem of describing unbounded σ-finite measures in semifinite algebras using von Neumann is similar to the Gleason problem.