  A measurable relation algebra is a relation algebra in which the identity
element is a sum of atoms that can be measured in the sense that the "size" of
each such atom can be defined in an intuitive and reasonable way (within the
framework of the first-order theory of relation algebras). A large class of
examples of such algebras, using systems of groups and coordinated systems of
isomorphisms between quotients of the groups, has been constructed. This class
of group relation algebras is not large enough to exhaust the class of all
measurable relation algebras.
  In the present article, the class of examples of measurable relation algebras
is considerably extended by adding one more ingredient to the mix: systems of
cosets that are used to "shift" the operation of relative multiplication. It is
shown that, under certain additional hypotheses on the system of cosets, each
such coset relation algebra with a shifted operation of relative multiplication
is an example of a measurable relation algebra. We also show that the class of
coset relation algebras does contain examples that are not representable as set
relation algebras. In a later article, it is shown that the class of coset
relation algebras is adequate to the task of describing all measurable relation
algebras in the sense that every atomic measurable relation algebra is
essentially isomorphic to a coset relation algebra.
