Let
(
X
,
B
)
(X,\mathcal {B})
be a standard Borel space,
R
⊂
X
×
X
R \subset X \times X
an equivalence relation
∈
B
×
B
\in \mathcal {B} \times \mathcal {B}
. Assume each equivalence class is countable. Theorem 1:
∃
\exists
a countable group G of Borel isomorphisms of
(
X
,
B
)
(X,\mathcal {B})
so that
R
=
{
(
x
,
g
x
)
:
g
∈
G
}
R = \{ (x,gx):g \in 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
α
,
β
\alpha ,\beta
be rationally independent irrationals on the circle
T
\mathbb {T}
, and f Borel:
T
→
T
\mathbb {T} \to \mathbb {T}
. Then
∃
\exists
Borel
g
,
h
:
T
→
T
g,h:\mathbb {T} \to \mathbb {T}
with
f
(
x
)
=
(
g
(
a
x
)
/
g
(
x
)
)
(
h
(
β
x
)
/
h
(
x
)
)
f(x) = (g(ax)/g(x))(h(\beta x)/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].