Precalibre pairs of measure algebras

AbstractWe consider Radon measures μ and pairs (κ,λ) of cardinals such that among every κ many positive measure sets there are λ many whose intersection is nonempty. Such families are connected with the cardinal invariants of the ideal of μ-null sets and have found applications in various subjects of topological measure theory. We survey many of such connections and applications and give some new ones. In particular we show that it is consistent to have a Corson compact space carrying a Radon measure of type c>ℵ1 and we partially answer a question of Haydon about measure precalibres

Universally Kuratowski-Ulam spaces and open-open games

We examine the class of spaces in which the second player has a winning strategy in the open--open game. We show that this spaces are not universally Kuratowski-Ulam. We also show that the games G and G7 introduced by P. Daniels, K. Kunen, H. Zhou [Fund. Math. 145 (1994), no. 3, 205--220] are not equivalent

Sequential closure in the space of measures

We show that there is a compact topological space carrying a measure which is not a weak* limit of finitely supported measures but is in the sequential closure of the set of such measures. We construct compact spaces with measures of arbitrarily high levels of complexity in this sequential hierarchy. It follows that there is a compact space in which the sequential closure cannot be obtained in countably many steps. However, we show that this is not the case for our spaces where the sequential closure is always obtained in countably many steps.Comment: (18 pages, a gap in an argument from the previous version fixed

Measure Recognition Problem

This is an article in mathematics, specifically in set theory. On the example of the Measure Recognition Problem (MRP) the article highlights the phenomenon of the utility of a multidisciplinary mathematical approach to a single mathematical problem, in particular the value of a set-theoretic analysis. MRP asks if for a given Boolean algebra \algB and a property $\Phi$ of measures one can recognize by purely combinatorial means if \algB supports a strictly positive measure with property $\Phi$. The most famous instance of this problem is MRP(countable additivity), and in the first part of the article we survey the known results on this and some other problems. We show how these results naturally lead to asking about two other specific instances of the problem MRP, namely MRP(nonatomic) and MRP(separable). Then we show how our recent work D\v zamonja and Plebanek (2006) gives an easy solution to the former of these problems, and gives some partial information about the latter. The long term goal of this line of research is to obtain a structure theory of Boolean algebras that support a finitely additive strictly positive measure, along the lines of Maharam theorem which gives such a structure theorem for measure algebras

Measurable centres in convolution semigroups

In a convolution semigroup over a locally compact group, measurability of the translation by a fixed element implies continuity. In other words, the measurable centre coincides with the topological centre.Comment: 8 page

Competition between standard and exotic double beta decays

We discuss the contributions of higher order terms in weak Hamiltonian to the standard two-neutrino double beta decay. The formalism for the unique first forbidden transitions has been developed, and it is shown that they can alter the two-electron energy spectrum. Yet, their effect is too small to screen the detection of exotic neutrinoless double beta decays, which are candidates for testing the physics beyond the standard model.Comment: 9 pages, latex, 1ps figures, minor changes, to appear in Phys. Lett.