A σ-ALGEBRA AND A CONCEPT OF LIMIT FOR BODIES

Recent developments in mechanics of continua (the search for optimal shapes of bodies, homogenization theory, the study of the trabecular structure of bones, the dynamics of immiscible mixtures, etc.) render some of the introductory axioms of continuum mechanics inadequate. Not only does one need to give meaning to the join and meet of two bodies, but also to extend the consequent algebra so as to encompass the result of a countable sequence of operations of join or meet; and one should also be able to define the limit of a sequence of bodies. To achieve this goal we propose here to define a body ab initio through the assignment of a probability measure dπ. We realize that π leaves, generally, too much of the texture of the body unspecified; to make up for this deficiency, we suggest the use of appropriate texture measures, reminescent of Tartar's H-measures.9