Extension of coherent lower previsions to unbounded random variables

We study the extension of coherent lower previsions from the set of bounded random variables to a larger set. An ad hoc method in the literature consists in approximating an unbounded random variable by a sequence of bounded ones. Its ‘extended’ lower prevision is then defined as the limit of the sequence of lower previsions of its approximations. We identify the random variables for which this limit does not depend on the details of the approximation, and call them previsible. We thus extend a lower prevision to previsible random variables, and we study the properties of this extension. We also consider the special case of super-modular lower previsions.