  Solving a well-known problem of Maharam, Talagrand [17] constructed an
exhaustive non uniformly exhaustive submeasure, thus also providing the first
example of a Maharam algebra that is not a measure algebra. To each exhaustive
submeasure one can canonically assign a certain countable ordinal, its
exhaustivity rank. In this paper, we use carefully constructed Schreier
families and norms derived from them to provide examples of exhaustive
submeasures of arbitrary high exhaustivity rank. This gives rise to uncountably
many non isomorphic separable atomless Maharam algebras.
