Interval-valued probability in the analysis of problems containing a mixture of possibilistic, probabilistic, and interval uncertainty

A simple definition of interval-valued probability measure (IVPM) is used and its implications are examined for problems in mathematical analysis. In particular, IVPMs are constructed and then used to develop the extension of these measures in such a way that probability, possibility, clouds, and intervals fit within the context of IVP. With the extension principle, integration and product measures that are derived below, mathematical analysis applied to this new structures is enabled. Optimization will be the mathematical analysis used to illustrate the approaches that are developed.