We define an analog of Voiculescu's free entropy for $n$-tuples of
unitaries $u\sb 1,\ldots u\sb n$ in a tracial von Neumann algebra $M$
which normalize a unital subalgebra $L\sp \infty[0,1]=B\subset
M$. Using this quantity, we define the free dimension $\delta\sb
0(u\sb 1,\ldots,u\sb n\between B)$. This number depends on $u\sb
1,\ldots u\sb n$ only up to orbit equivalence over $B$. In particular,
if $R$ is a measurable equivalence relation on $[0,1]$ generated by
$n$ automorphisms $\alpha\sb 1,\ldots \alpha\sb n$, let $u\sb 1,\ldots
u\sb n$ be the unitaries implementing $\alpha\sb 1,\ldots \alpha\sb n$
in the Feldman-Moore crossed product algebra $M=W\sp
\ast([0,1],R)\supset B=L\sp \infty[0,1]$. Then the number
$\delta(R)=\delta\sb 0(u\sb 1,\ldots u\sb n\between B)$ is an
invariant of the equivalence relation $R$. If $R$ is treeable,
$\delta(R)$ coincides with the cost $C(R)$ of $R$ in the sense of
D. Gaboriau. In particular, it is $n$ for an equivalence relation
induced by a free action of the free group $\mathbb {F}\sb n$. For a
general equivalence relation $R$ possessing a finite graphing of
finite cost, $\delta(R)\leq C(R)$. Using the notion of free dimension,
we define a dynamical entropy invariant for an automorphism of a
measurable equivalence relation (or, more generally, of an
$r$-discrete measure groupoid) and give examples.