1 · Check the parts of a measure rule

A proposed rule needs a condensed ring and a module of allowed measures for every extremally disconnected probe. Point masses must be included, and separate pieces must receive independent data.

Try this. Turn each requirement off and on. Without Dirac measures, points cannot enter the measure module. Without the disjoint-union rule, two separate pieces cannot be handled independently.

Definition 7.1 lists these data. Definition 7.4 adds the derived condition that defines an analytic ring. The controls check necessary inputs but cannot prove that final condition.