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In this paper a basic theory is
developed for inverse (or projective) systems of group-valued measures. This theory
parallels the one for nonnegative measures. However, many of the results are new
even in the real case.
The main tools for dealing with group-valued measures are the concepts and
results given by Sion in “Outer measures with values in a topological group”, Proc.
London Math. Soc., 19 (1969), 89-106. When dealing with inverse systems
the point of view adopted is that of Mallory and Sion, “Limits of inverse
systems of measures”. Ann. Inst. Fourier, Tome 21, Fasc. 1 (1971) 25-57. This
viewpoint involves finding a limit measure first on a large space Λ and then
studying conditions under which this will yield a limit measure on some
subset of Λ. By introducing the concept of almost-sequential maximality,
this paper not only extends known results but is also able to indicate a
connection between “abstract” and “topological” methods for producing a limit
measure.
In the last section the results obtained are applied to cylinder measures. Here
again the viewpoint adopted differs somewhat from the usual one, even for
nonnegative measures, and enables one to study a variety of possibilities for a target
space on which to place a limit measure.
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