The subset of lattice regular zero-one valued measures on an algebra generated by a lattice (a Wallman-type space) which integrates all lattice continuous functions on an arbitrary setX is introduced and some properties of it are presented.
We study the doubling property of binomial measures on the middle interval Cantor set. We obtain a necessary and sufficient condition that enables a binomial measure to be doubling. Then we determine those doubling binomial measures which can be extended to be doubling on [0,1]. Finally, we construct a compact set X in [0,1] and a doubling measure μ on X, such that and is doubling on EX, where EX is the set of accumulation points of X and FX is the set of isolated points of X.
The notions of parallel sum and parallel difference of two nonnegative forms were introduced and studied by Hassi, Sebestyén, and de Snoo in  and . In this paper we consider the parallel subtraction with much circumstances. Criteria are established for the solvability of the equation with an unknown when and are given. We identify as the minimal solution, and characterize all the solutions under the assumption where λ>1. The Galois correspondence induced by the map is also studied. We show that if the equation is solvable, then there is a unique -closed solution, namely . Finally, we consider some extremal problems such as the extreme points of the interval , and the characterization of the minimal forms in terms of the parallel sum.