Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Fundamenta Mathematicae / Online First articles

Abstract

Grade-$d$ measures on a $\sigma $-algebra $\mathcal {A}\subseteq 2^X$ 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 $\lambda $ on $(X,\mathcal {A})^d$ produces a grade-$d$ measure as its diagonal $\widetilde{\lambda}(A):=\lambda (A,\ldots ,A)$, and we prove that as soon as $d\ge 2$, measures (as opposed to polymeasures) do not suffice: the separate $\sigma $-additivity of a $\lambda $ producing $\mu =\widetilde{\lambda}$ cannot, generally, be amplified to global $\sigma $-additivity. This amends a result in the literature, asserting the contrary in case $d=2$.