  Let G be an affine algebraic group and let X be an affine algebraic variety.
An action $G\times X \to X$ is called observable if for any G-invariant,
proper, closed subset Y of X there is a nonzero invariant $f\in K[X]^G$ such
that f(Y) =0. We characterize this condition geometrically as follows. The
action $G\times X \to X$ is observable if and only if (1) there is a nonempty
open subset $U\subseteq X$ consisting of closed orbits, and (2) the field
$K(X)^G$ of G-invariant rational functions on X is equal to the quotient field
of $K[X]^G$. In case G is reductive, we conclude that there exists a unique,
maximal, G-stable, closed subset $X_{\soc}$ of $X$ such that $G\times X_{\soc}
\to X_{\soc}$ is observable. Furthermore, the canonical map $X_{\soc}// G \to
X//G$ is finite and bijective.
