Algebraic theory of measure algebras

Abstract

A. Horn and A. Tarski initiated the abstract theory of measure algebras. Independently V. Sushchansky, B. Oliynyk and P. Cameron studied the direct limits of Hamming spaces. In the current paper, we introduce new examples of locally standard measure algebras and complete the classification of countable locally standard measure algebras. Countable unital locally standard measure algebras are in one-to-one correspondence with Steinitz numbers. Given a Steinitz number s such measure algebra is isomorphic to the Boolean algebra of s-periodic sequences of 0 and 1. Nonunital locally standard measure algebras are parametrized by pairs (s, r), where s is a Steinitz number and r is a real number greater or equal to 1. We also show that an arbitrary (not necessarily locally standard) measure algebra is embeddable in a metric ultraproduct of standard Hamming spaces. In other words, an arbitrary measure algebra is sofic.