Relationships between the processes of integration in two measure spaces connected by a transformation have been investigated under a variety of conditions. The role of the transformation in these studies is specified by weight functions. Under weak restrictions it has been observed that these weight functions possess Lebesgue type decompositions [4]. The purpose of this paper is to find conditions under which signed weight functions possess Jordan decompositions. These results answer some of the questions posed by Craft [7] in a much more restricted setting. In a subsequent paper Jordan decompositions will be used to develop transformation formulas and chain rules under restrictions differing from those in the literature (see references).