Measures on orthoideals and L <inf>1</inf>-spaces associated with positive operators

© 2016, Pleiades Publishing, Ltd.We study measures on orthoideals in the set of projections in von Neumann algebra, which appear within a framework of L1-spaces associated with positive operators in the algebra. We show that there exists a positive measure on orthoideal, which cannot be extended to a normal weight on the algebra, but can be represented as a difference of two positive measures extendable to normal weights