We show that the minimal cardinality of a dense subset of the measure algebra is the same as the minimal cardinality of a base of the ideal of Lebesgue measure zero subsets of the real line. 0. Introduction. Let (F, <) be a given partial ordering. A subset D ç P is called dense if for any p g F there exists d g D such that d < p. A subset D is called upward dense if D is dense in (F, > ). Let A...

The Open Colouring Axiom implies that the measure algebra cannot be embedded into P(N)/fin. We also discuss errors in previous results on the embeddability of the measure algebra. Introduction. The aim of this paper is to prove the following result. Main Theorem. The Open Colouring Axiom implies that the measure algebra cannot be embedded into the Boolean algebra P(N)/fin. By “the measure algeb...

We introduce the notion of Boolean measure algebra. It can be described shortly using some standard notations and terminology. If B is any Boolean algebra, let B N denote the algebra of sequences (xn); xn 2 B: Let us write p k 2 B N the sequence such that p k (i) = 1 if i k and p k (i) = 0 if k < i: If x 2 B, denote by x 2 B N the constant sequence x = (x; x; x; : : :): We deene a Boolean measu...

Here, instead of taking a σ-algebra as given and then defining a measure on this σ-algebra (namely, on the measurable sets), we take an outer measure as given and then define measurable sets using this outer measure. Carathéodory’s theorem states that the collection M of ν-measurable sets is a σ-algebra and that the restriction of ν to M is a complete measure. Suppose that (X, ρ) is a metric sp...

Endomorphisms of the measure algebra commutative hypergroups are investigated. We focus on derivations and higher order which closely related to moment function sequences rank. describe exact connection between those endomorphisms if it is considered as a module over ring continuous functions.

This note being devoted to some aspects of the inverse problem of representation theory explicates the links between researches on the Sklyanin algebras and the author’s (based on the noncommutative geometry) approach to the setting free of hidden symmetries in terms of ”the quantization of constants”. Namely, the Racah–Wigner algebra for the Sklyanin algebra is constructed. It may be considere...

We present necessary and sufficient conditions for the existence of a countably additive measure on a Boolean σ-algebra. For instance, a Boolean σ-algebra B is a measure algebra if and only if B−{0} is the union of a chain of sets C1 ⊂ C2 ⊂ ... such that for every n, (i) every antichain in Cn has at mostK(n) elements (for some integerK(n)), (ii) if {an}n is a sequence with an / ∈ Cn for each n,...

We prove that a completely symmetric and trace-free rank-4 tensor is, up to sign, a Bel-Robinson type tensor, i.e., the superenergy tensor of a tensor with the same algebraic symmetries as the Weyl tensor, if and only if it satisfies a certain quadratic identity. This may be seen as the first Rainich theory result for rank-4 tensors.