From the summary and the introduction: This paper introduces and studies a generalization of the notion of martingale which allows for a generalization of the concept of a Markov chain and a generalization of the concept of harmonic and superharmonic functions. The theory is supported by examples and techniques that suggest the natural character of the material developed. As the paper proceeds, there will be many supplementary lines of interest which will capture the reader's interest. We hope that among these will be further considerations of metrics on sub sigma fields and a further exploration of pointwise convergence results for processes suggested by weak martingales. Finally, it is important to emphasize that this paper serves only as an introduction to a body of material. There will certainly be as many questions generated for further discussion as there are answered and therein will lie the real excitement.