The optimal differentiation basis and liftings of 𝐿^{∞}

Bliedtner, Jürgen; Loeb, Peter

doi:10.1090/S0002-9947-00-02615-5

The optimal differentiation basis and liftings of $L^{{\infty }}$
HTML articles powered by AMS MathViewer

by Jürgen Bliedtner and Peter A. Loeb PDF

Trans. Amer. Math. Soc. 352 (2000), 4693-4710 Request permission

Abstract:

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.

References

J. Bliedtner and W. Hansen, Potential theory, Universitext, Springer-Verlag, Berlin, 1986. An analytic and probabilistic approach to balayage. MR 850715, DOI 10.1007/978-3-642-71131-2
Jürgen Bliedtner and Peter A. Loeb, A measure-theoretic boundary limit theorem, Arch. Math. (Basel) 43 (1984), no. 4, 373–376. MR 802315, DOI 10.1007/BF01196663
J. Bliedtner and P. Loeb, A reduction technique for limit theorems in analysis and probability theory, Ark. Mat. 30 (1992), no. 1, 25–43. MR 1171093, DOI 10.1007/BF02384860
Jürgen Bliedtner and Peter A. Loeb, Best filters for the general Fatou boundary limit theorem, Proc. Amer. Math. Soc. 123 (1995), no. 2, 459–463. MR 1219720, DOI 10.1090/S0002-9939-1995-1219720-2
____________, Sturdy harmonic functions and their integral representations, preprint.
P. Erdös and T. Grünwald, On polynomials with only real roots, Ann. of Math. (2) 40 (1939), 537–548. MR 7, DOI 10.2307/1968938
Bernd Eifrig, Ein nicht-standard Beweis für die Existenz eines starken Lifting in ${\cal L}_{\infty }\langle 0,1] w\rangle$, Contributions to non-standard analysis (Sympos., Oberwolfach, 1970), Studies in Logic and Foundations of Math., Vol. 69, North-Holland, Amsterdam, 1972, pp. 81–83 (German). MR 0583871
B. Eifrig, Ein Nicht-Standard-Bewels für die Existenz eines Liftings, Measure theory (Proc. Conf., Oberwolfach, 1975) Lecture Notes in Math., Vol. 541, Springer, Berlin, 1976, pp. 133–135 (German). MR 0442679
Gerald B. Folland, Real analysis, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. Modern techniques and their applications; A Wiley-Interscience Publication. MR 767633
Bent Fuglede, Remarks on fine continuity and the base operation in potential theory, Math. Ann. 210 (1974), 207–212. MR 357826, DOI 10.1007/BF01350584
Zoltán Füredi and Peter A. Loeb, On the best constant for the Besicovitch covering theorem, Proc. Amer. Math. Soc. 121 (1994), no. 4, 1063–1073. MR 1249875, DOI 10.1090/S0002-9939-1994-1249875-4
Siegfried Graf and Heinrich von Weizsäcker, On the existence of lower densities in noncomplete measure spaces, Measure theory (Proc. Conf., Oberwolfach, 1975) Lecture Notes in Math., Vol. 541, Springer, Berlin, 1976, pp. 155–158. MR 0450971
Albert E. Hurd and Peter A. Loeb, An introduction to nonstandard real analysis, Pure and Applied Mathematics, vol. 118, Academic Press, Inc., Orlando, FL, 1985. MR 806135
A. Ionescu Tulcea and C. Ionescu Tulcea, Topics in the theory of lifting, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 48, Springer-Verlag New York, Inc., New York, 1969. MR 0276438, DOI 10.1007/978-3-642-88507-5
P. A. Loeb, Opening the covering theorems of Besicovitch and Morse, Mathematica Moravica, Special Volume 1997, Proceedings of the 1995 International Workshop in Analysis and its Applications, 3–11.
Jaroslav Lukeš, Jan Malý, and Luděk Zajíček, Fine topology methods in real analysis and potential theory, Lecture Notes in Mathematics, vol. 1189, Springer-Verlag, Berlin, 1986. MR 861411, DOI 10.1007/BFb0075894
W. A. J. Luxemburg, A remark on the Cantor-Lebesgue lemma, Contributions to non-standard analysis (Sympos., Oberwolfach, 1970), Studies in Logic and Found. Math., Vol. 69, North-Holland, Amsterdam, 1972, pp. 41–46. MR 0470593
Frank Wattenberg, Nonstandard measure theory: avoiding pathological sets, Trans. Amer. Math. Soc. 250 (1979), 357–368. MR 530061, DOI 10.1090/S0002-9947-1979-0530061-4