  The Whittle index, which characterizes optimal policies for controlling
certain single restless bandit projects (a Markov decision process with two
actions: active and passive) is the basis for a widely used heuristic index
policy for the intractable restless multiarmed bandit problem. Yet two
roadblocks need to be overcome to apply such a policy: the individual projects
in the model at hand must be shown to be indexable, so that they possess a
Whittle index; and the index must be evaluated. Such roadblocks can be
especially vexing when project state spaces are real intervals, as in recent
sensor scheduling applications. This paper presents sufficient conditions for
indexability (relative to a generalized Whittle index) of general real-state
discrete-time restless bandits under the discounted criterion, which are not
based on elucidating properties of the optimal value function and do not
require proving beforehand optimality of threshold policies as in prevailing
approaches. The main contribution is a verification theorem establishing that,
if project performance metrics under threshold policies and an explicitly
defined marginal productivity (MP) index satisfy three conditions, then the
project is indexable with its generalized Whittle index being given by the MP
index, and threshold policies are optimal for dynamic project control.
