C.C. Moore’s scientific contributions

Let (X, B) be a standard Borel space, R Ì X x X an equivalence. relation £& x Assume each equivalence class is countable. Theorem 1:3 a countable group G of Borel isomorphisms of (X, $) so that R - {(*, gx): g Î G. G is far from unique. However, notions like invariance and quasi-invariance and R-N derivatives of measures depend only on R, not the choice of G. We develop some of the ideas of Dye [1], [2] and Krieger [1]—[5] in a fashion explicitly avoiding any choice of G; we also show the connection with virtual groups. A notion of “module over R” is defined, and we axiomatize and develop a cohomology theory for R with coefficients in such a module. Surprising application (contained in Theorem 7): let a, p be rationally independent irrationals on the circle T, and /Borel: T->T. Then 3 Borel g, h: T-*T with f(x) - (g(ax)/g(x))(h(bx)/h(x)) a.e. The notion of “skew product action*' is generalized to our context, and provides a setting for a generalization of the Krieger invariant for the R-N derivative of an ergodic transformation: we define, for a cocycle c on R with values in the group A, a subgroup of A depending only on the cohomology class of c, and in Theorem 8 identify this with another subgroup, the “normalized proper range” of c, defined in terms of the skew action. See also Schmidt [1].