  A relation algebra is called measurable when its identity is the sum of
measurable atoms, and an atom is called measurable if its square is the sum of
functional elements. In this paper we show that atomic measurable relation
algebras have rather strong structural properties: they are constructed from
systems of groups, coordinated systems of isomorphisms between quotients of the
groups, and systems of cosets that are used to "shift" the operation of
relative multiplication. An atomic and complete measurable relation algebra is
completely representable if and only if there is a stronger coordination
between these isomorphisms induced by a scaffold (the shifting cosets are not
needed in this case). We also prove that a measurable relation algebra in which
the associated groups are all finite is atomic.
