Applications of fuzzy set theory and near vector spaces to functional analysis

We prove an original version of the Hahn-Banach theorem in the fuzzy setting. Convex compact sets occur naturally in set-valued analysis. A question that has not been satisfactorily dealt with in the literature is: What is the relationship between collections of such sets and vector spaces? We thoroughly clarify this situation by making use of R°adstr ¨om’s embedding theorem, leading up to the definition of a near vector space. We then go on to successfully apply these results to provide an original method of proof of Doob’s decomposition of submartingales

Doob’s decomposition of set-valued submartingales via ordered near vector spaces

We use ideas from measure-free martingale theory and R˚adstr¨om’ completion of a near vector space to derive a Doob decomposition of submartingales in ordered near vector spaces. As a special cases thereof, we obtain the Doob decomposition of set-valued submartingales, as noted by Daures, Ni and Zhang, and an analogue of the Doob decomposition of set-valued submartingales, as noted by Shen and Wang.Quaestiones Mathematicae 32(2009), 247–26