We investigate complete Boolean algebras that carry a continuous submeasure.

Let B be a complete ccc Boolean algebra and let τs be the topology on B induced by the algebraic convergence of sequences in B. 1. Either there exists a Maharam submeasure on B or every nonempty open set in (B, τs) is topologically dense. 2. It is consistent that every weakly distributive complete ccc Boolean algebra carries a strictly positive Maharam submeasure. 3. The topological space (B, τ...

A submeasure θ on a Boolean algebra B is pathological if it does not dominate a positive finitely additive functional. It is ε-exhaustive if every sequence {An} of pairwise disjoint sets in B satisfies lim supn θ(An) ≤ ε. It is exhaustive if it is 0-exhaustive. It is normalized if θ(1B) = 1. Question 1. Is there a normalized exhaustive pathological submeasure on the algebra of clopen subsets of...

Exhibiting a new type of measure concentration, we prove uniform concentration bounds for measurable Lipschitz functions on product spaces, where is taken with respect to the metric induced by weighted covering index set product. Our proof combines Herbst argument an analogue Shearer's lemma differential entropy. We give quantitative “geometric” classification diffuse submeasures into elliptic,...

Let B be a complete ccc Boolean algebra and let τs be the topology on B induced by the algebraic convergence of sequences in B. 1. Either there exists a Maharam submeasure on B or every nonempty open set in (B, τs) is topologically dense. 2. It is consistent that every weakly distributive complete ccc Boolean algebra carries a strictly positive Maharam submeasure. 3. The topological space (B, τ...

In this paper we study a generalization of a submeasure notion which is related to a probabilistic concept, especially to Menger spaces where triangular norms play a crucial role. The resulting notion of a τT -submeasure is suitable for modeling those situations in which we have only probabilistic information about the measure of the set. We characterize a class of universal τT -submeasures (i....

A Boolean algebra carries a strictly positive exhaustive submeasure if and only if it has a sequential topology that is uniformly Fréchet.

We construct an exhaustive submeasure that is not equivalent to a measure. This solves problems of J. von Neumann (1937) and D. Maharam (1947).

Every uniformly exhaustive submeasure is equivalent to a measure. From this, we deduce that every vector measure with compact range in an F-space has a control measure. We also show that co (or any E.-space) is a T;space, i.e. cannot be realized as the quotient of a nonlocally convex F-space by a one-dimensional subspace.

In our earlier paper (Arch. Math. 91 (2008), 76–85), we proved that if F is a sequence of finite nonempty subsets of N such that a certain quantity t(F ) is finite, then the associated submeasure dF on N is nonatomic. In the present note, we give two curious characterizations of the set of such F ’s. Mathematics Subject Classification (2000). 05A17, 11B05, 28A12.