Abstract
A representation of the projective system of abstract σ-finite measures on a topological family is given and with it a general characterization of their projective limits is obtained. Strong and weak direct limits of direct systems of measures as well as the duality between them are characterized with detailed analysis. This is used to prove several results of both theoretical and applicational importance. These include obtaining the equivalence of regular martingales and some projective systems admitting limits, measure representations of general semi-martingales, an extension theorem of product conditional measures, and a generalization of Rokhlin's theorem on completely positive entropy sequences of Lebesgue systems to general probability spaces. Further characterizations of projective and direct limits receive an extended treatment, indicating a great potential for future works.
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