Related papers: On submeasures on Boolean algebras
It is consistent that every weakly distributive complete ccc Boolean algebra carries a strictly positive Maharam submeasure.
The paper investigates possible generalisations of Maharam's theorem to a classification of Boolean algebras that support a finitely additive measure. We prove that Boolean algebras that support a finitely additive non-atomic uniformly…
We present a necessary and sufficient condition for a Boolean algebra to carry a finitely additive measure.
If I is a suitably definable sigma-ideal on the real line and the factor algebra of Borel sets modulo I is weakly distributive then the algebra carries a Maharam submeasure.
A Boolean algebra carries a strictly positive exhaustive submeasure if and only if it has a sequential topology that is uniformly Frechet.
Sachs showed that a Boolean algebra is determined by its lattice of subalgebras. We establish the corresponding result for orthomodular lattices. We show that an orthomodular lattice L is determined by its lattice of subalgebras Sub(L), as…
In this paper we discuss the existence of a control measure for a family of measures on a Boolean algebra. We obtain a necessary and sufficient condition and several related results, including a new criterion for weak compactness for…
Solving a well-known problem of Maharam, Talagrand [17] constructed an exhaustive non uniformly exhaustive submeasure, thus also providing the first example of a Maharam algebra that is not a measure algebra. To each exhaustive submeasure…
We give a fresh account of the astonishing interplay between abelian von Neumann algebras, L^\infty-spaces and measure algebras, including an exposition of Maharam's theorem from the von Neumann algebra perspective.
We prove a Theorem about the relationship between the Depth of the ultraproduct of Boolean algebras, divided by an ultrafilter, and the products of the depths of each component. This answers (partly) an open problem of Monk.
We construct Boolean Algebras answering questions of Monk on cardinal invariants. The results are proved in ZFC (rather than giving consistency results). We deal with the existence of superatomic Boolean Algebras with ``few automorphisms'',…
The original theme of the paper is the existence proof of ``there is < eta_alpha : alpha < lambda > which is a (lambda,J)-sequence for < I_i:i<delta >, a sequence of ideals. This can be thought of as in a generalization to Luzin sets and…
This note is concerned with an extension, at second order, of an inequality on the discrete cube $C_n=\{-1,1\}$ (equipped with the uniform measure) due to Talagrand (\cite{TalL1L2}). As an application, the main result of this note is a…
We discuss new problems in universal algebraic geometry and explain them by boolean equations.
We present necessary and sufficient conditions for the existence of a countably additive measure on a complete Boolean algebra.
Scaled Boolean algebras are a category of mathematical objects that arose from attempts to understand why the conventional rules of probability should hold when probabilities are construed, not as frequencies or proportions or the like, but…
We develop a direct method to recover an orthoalgebra from its poset of Boolean subalgebras. For this a new notion of direction is introduced. Directions are also used to characterize in purely order-theoretic terms those posets that are…
We construct an exhaustive submeasure that is not equivalent to a measure. This solves problems of J. von Neumann (1937) and D. Maharam (1947).
A Boolean $\sigma$-algebra $B$ is a measure algebra if and only if it is weakly distributive and uniformly concentrated.
We consider the problem of constructing semisimple subalgebras of real (semi-) simple Lie algebras. We develop computational methods that help to deal with this problem. Our methods boil down to solving a set of polynomial equations. In…