Observable actions of algebraic groups

Let G be an affine algebraic group and let X be an affine algebraic variety. An action G × X → X is called observable if for any G-invariant, proper, closed subset Y of X there is a nonzero invariant f ∈ \Bbbk\Bbbk X] ^G such that f| _Y = 0. We characterize this condition geometrically as follows. The action G × X → X is observable if and only if:

	(1) the action is stable, that is there exists a nonempty open subset U ⊆ X consisting of closed orbits; and
	(2) the field \Bbbk\Bbbk(X) G of G-invariant rational functions on X is equal to the quotient field of \Bbbk\BbbkX] G .