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Transactions of the American Mathematical Society

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The optimal differentiation basis and liftings of $L^{{\infty}}$

Authors: Jürgen Bliedtner and Peter A. Loeb
Journal: Trans. Amer. Math. Soc. 352 (2000), 4693-4710
MSC (1991): Primary 28A15; Secondary 28A51, 26E35, 31C40, 54B99.
Published electronically: April 21, 2000
MathSciNet review: 1709771
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There is an optimal way to differentiate measures when given a consistent choice of where zero limits must occur. The appropriate differentiation basis is formed following the pattern of an earlier construction by the authors of an optimal approach system for producing boundary limits in potential theory. Applications include the existence of Lebesgue points, approximate continuity, and liftings for the space of bounded measurable functions - all aspects of the fact that for every point outside a set of measure $0$, a given integrable function has small variation on a set that is ``big'' near the point. This fact is illuminated here by the replacement of each measurable set with the collection of points where the set is ``big'', using a classical base operator. Properties of such operators and of the topologies they generate, e.g., the density and fine topologies, are recalled and extended along the way. Topological considerations are simplified using an extension of base operators from algebras of sets on which they are initially defined to the full power set of the underlying space.

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Additional Information

Jürgen Bliedtner
Affiliation: Fachbereich Mathematik, Universität Frankfurt, Robert-Mayer-Str. 6-8, D-60054, Frankfurt/M, Germany

Peter A. Loeb
Affiliation: Department of Mathematics, University of Illinois, 1409 West Green St., Urbana, Illinois 61801

Keywords: Differentiation basis, Lebesgue point, base operator, density topology, monad, lifting.
Received by editor(s): April 2, 1998
Received by editor(s) in revised form: April 23, 1999
Published electronically: April 21, 2000
Additional Notes: The work of both authors was supported in part by NSF Grant DMS 96-22474.
Article copyright: © Copyright 2000 American Mathematical Society