Approximate homomorphisms

Abstract

Every mapping between finite Boolean algebras which is approximately a homomorphism with respect to some measure φ on the range (see Definition 2) can be approximated by a homomorphisni within a constant error. An analogous statement fails in case when φ is a pathological submeasure. The key to our proof is the fact that a subset of a finite Boolean algebra {0, 1}[m] which "almost everywhere" looks like an ultrafilter has to be close to some fixed ultrafilter.