Mass continuous cochains are differential forms

Abstract

for each o-E CkRn. It is natural to seek conditions on an arbitrary cochain X sufficient to insure the existence of such a corresponding differential form. Such conditions have usually been expressed in terms of continuity hypotheses on X. In [5] it was shown that to each flat cochain and to each sharp cochain there corresponds such a differential form. In [4] it was shown that to each local L1 1-cochain in the plane there corresponds such a 1-fortn. In this paper we make the observation that the existence of these differential fortns as well as the form indicated in the example above follows almost immediately from the assumption of the well otdering principle and the continuum hypothesis. This observation is, in fact, an easy consequence of H. Federer's remarks on the sigma-finite hypothesis of the Radon-Nikodym theorem which will appear in his book on geometric measure theory. It seems useful, however, to make these