The Filter Dichotomy and medial limits

The Filter Dichotomy says that every uniform nonmeager filter on the integers is mapped by a finite-to-one function to an ultrafilter. The consistency of this principle was proved by Blass and Laflamme. A medial limit is a universally measurable function from P(ω) to the unit interval [0, 1] which is finitely additive for disjoint sets, and maps singletons to 0 and ω to 1. Christensen and Mokobodzki independently showed that the Continuum Hypothesis implies the existence of medial limits. We show that the Filter Dichotomy implies that there are no medial limits. 2000 Mathematics Subject Classification 03E35; 28A20 1 Universally measurable sets A measure on a set X is a function μ whose domain is some σ-algebra of subsets of X, with codomain [0,∞], such that μ is countably additive for disjoint families (by default, measures are countably additive, but we deal also with finitely additive measures, which are finitely additive but not necessarily countably additive). A set is said to be measurable with respect to μ if it is in the domain of μ. A Borel measure is a measure on a topological space whose domain contains the Borel subsets of the space. A measure is complete if all subsets of sets of measure 0 are in the domain of the measure (and thus have measure 0). The completion of a measure is the smallest complete measure extending it. If μ is a Borel measure on a topological space X, and μ∗ is the completion of μ, then a set A ⊆ X is in the domain of μ∗ if and only if there is a set B in the domain of μ such that the symmetric difference A△B is contained in a set of μ-measure 0 (see 212C of [8]). A measure μ on a set X is a probability measure if μ(X) = 1, finite if μ(X) is finite, σ-finite if X is a countable union of sets of finite measure, and atomless if singletons have measure 0. A subset of a topological space is said to be universally measurable if it is measurable with respect to every complete σ-finite Borel measure on the space (see [15, 21], and 434D of [9], for instance). The collection of universally measurable sets does not change if one replaces “σ-finite” with “finite” or requires the measures to be atomless (see 211X(e) of [8]). ∗Supported in part by NSF grant DMS-0801009.