  A function $J$ defined on a family $C$ of stationary processes is finitely
observable if there is a sequence of functions $s_n$ such that $s_n(x_1 ...
x_n)\to J(X)$ in probability for every process $X=(x_n)\in C$. Recently,
Ornstein and Weiss roved the striking result that if $C$ is the class of
aperiodic ergodic finite valued processes, then the only finitely observable
isomorphism invariant on $C$ is entropy. We sharpen this in several ways. Our
main theorem is that if $X \to Y$ is a zero-entropy extension of finite entropy
ergodic systems and $C$ is the family of processes arising from $X$ and $Y$,
then every finitely observable function on $C$ is constant. This implies
Ornstein and Weiss' result, and extends it to many other families of processes,
e.g. it shows that there are no nontrivial finitely observable isomorphism
invariants for processes arising from Kronecker systems, mild and strong mixing
zero entropy systems. It also implies that any finitely observable isomorphism
invariant defined on the family of processes arising from irrational rotations
must be constant for rotations belonging to a set of full Lebesgue measure.
