Non-representable quantum measures

Grade-d measures on a σ-algebra A⊆2X over a set X are generalizations of measures satisfying one of a hierarchy of weak additivity-type conditions initially introduced as interference operators in quantum mechanics. Every signed polymeasure λ on (X,A)d produces a grade-d measure as its diagonal λ˜(A):=λ(A,…,A), and we prove that as soon as d≥2, measures (as opposed to polymeasures) do not suffice: the separate σ-additivity of a λ producing μ=λ˜ cannot, generally, be amplified to global σ-additivity. This amends a result in the literature, asserting the contrary in case d=2.