Continuity of the integral as a function of the domain

This paper proves an isomorphism theorem for cochains and differential forms, before passing to cohomology. De Rham’s theorem is a consequence. This leads to an extension of much of calculus and homology theory to nonsmooth domains, called chainlets,and makes available combinatorial techniques for smooth domains that limit to the classic analytic methods. We find maximal subspaces of L ¹ forms that satisfy Stokes’s theorem for domains of chainlets giving a measurable, as well as optimal, extension of the theory.