Measure recognition problem.

This is a paper in mathematics, specifically in set theory. On the example of the measure recognition problem (MRP), the paper 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, B, and a property, Phi, of measures, one can recognize by purely combinatorial means if B 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 paper, 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 (non-atomic) and MRP (separable). Then, we show how our recent work gives an easy solution to the former of these problems and 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.