Vol. 44, No. 2, 1973

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Inverse systems of group-valued measures

H. Millington and Maurice Sion

Vol. 44 (1973), No. 2, 637–650

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.

Mathematical Subject Classification
Primary: 28A45
Received: 9 March 1971
Published: 1 February 1973
H. Millington
Maurice Sion