The measure algebra does not always embed

Abstract

The Open Colouring Axiom implies that the measure algebra cannot be embedded into P(N)/fin. We also discuss errors in previous results on the embeddability of the measure algebra. Introduction. The aim of this paper is to prove the following result. Main Theorem. The Open Colouring Axiom implies that the measure algebra cannot be embedded into the Boolean algebra P(N)/fin. By "the measure algebra" we mean the quotient of the σ-algebra of Borel sets of the real line by the ideal of sets of measure zero. There are various reasons, besides sheer curiosity, why it is of interest to know whether the measure algebra can be embedded into P(N)/fin. One reason is that there is great interest in determining what the subalgebras of P(N)/fin are. One of the earliest and most influential results in this direction is Parovichenko's theorem from [14], which states that every Boolean algebra of size ℵ 1 can be embedded into P(N)/fin, with the obvious corollary that the Continuum Hypothesis (CH) implies that P(N)/fin is a universal Boolean algebra of size c: a Boolean algebra embeds into P(N)/fin iff it is of size c or less. It is therefore natural to ask how much of this universality remains without assumptions beyond ZFC. It has long been known that every σ-centered Boolean algebra embeds into P(N)/fin but the question for more general c.c.c. Boolean algebras has proven to be much more difficult-with the case of the measure algebra being seen as a touchstone.

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