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Limits of inverse systems of measures Mallory, Donald James

Abstract

In this paper we are concerned with the problem of finding 'limits' of inverse (or projective) systems of measure spaces (for a definition of these see e.g. Choksi: Inverse Limits of Measure, Spaces, Proc. London Math. Soc. 8, 1958). Our basic limit measure, ῦ, is placed on the Cartesian product of the spaces instead of on the inverse limit set, L. As a result we obtain an existence theorem for this measure with fewer conditions on the system than are usually needed. We also investigate the existence of a limit measure on L by restricting our measure ῦ to L. This enables us to generalize known results and to explain some of the difficulties encountered by the standard inverse limit measure. In particular we show that the product topology may be too fine to allow the limit measure to have good topological properties' (e.g. to be Radón). Another topology which is related to the product structure is introduced and we show that limit measures which are Radón w.r.t. this topology can be obtained for a wide class of inverse systems of measure spaces.

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