  Whereas "convexification by aggregation" is a well-understood procedure in
mathematical economics, "convexification by randomization" has largely been
limited to theories of statistical decision-making, optimal control and
non-cooperative games. In this paper, in the context of classical Walrasian
general equilibrium theory, we offer a comprehensive treatment of {\it relaxed
economies} and their {\it relaxed Walrasian equilibria}: our results pertain to
a setting with a finite or a continuum of agents, and a continuum of
commodities modeled either as an ordered separable Banach space or as an
$L^\infty$-space. As a substantive consequence, we demonstrate that the
convexity hypothesis can be removed from the original large economy under the
saturation hypothesis, and that existing results in the antecedent literature
can be effortlessly recovered.
