# Peter Loeb

Department of Mathematics
University of Illinois at Urbana-Champaign
1409 W. Green Street (MC-382)
Urbana, Illinois 61801-2975

Office: 350 Altgeld Hall
Phone: (217) 333-8664
FAX:  (217) 333-9576
e-mail: PeterA3.aol.com

Teaching: Fall 2010 Semester, Math 542 MWF 11. Office hours right after class.

Research:

Peter Loeb's research is centered in problems of representing measures and ideal boundaries in potential theory and also the application of nonstandard models to mathematical analysis. His research often uses measure spaces that are called "Loeb spaces" in the literature. Loeb and other researchers have used these methods in areas such as potential theory, stochastic processes, mathematical economics, and mathematical physics. There are many results in pure and applied mathematics, such as found in Yeneng Sun's work on the law of large numbers and Loeb's recent work with Sun on the purification of measure-valued maps, that are only valid when the underlying parameter space is a Loeb space.

Another major developed in Loeb's work is a method for showing that a limit of ratios yields a Radon-Nikodym derivative. This technique casts new light on the martingale convergence theorem and significantly simplifies the treatment of fine limit theorems in potential theory and measure differentiation theorems. An important consequence is the existence of simple boundary approach neighborhood systems in probability and potential theory that, after fixing a suitable normalization, are the "best" possible in terms of producing Radon-Nikodym derivatives as limits at the boundary. The existence of such "best" systems had not been known even for harmonic functions on the unit disk, and a result of Doob suggested that no such system could exist. An analogous construction for the differentiation of measures has also been published. Each measurable set acts as a functional on measures: The value of the functional is the measure of the set. The larger the collection of sets one has at any stage of the filtration process associated with a point, the more information one obtains when a limit exists. Given a suitable normalization, there is an "optimal", i.e. coarsest filtration process that can be used to differentiate measures. Applications include the existence of Lebesgue points and liftings of L^{âˆž}. New results on base operators and topologies are part of this research.

Related to the differentiation of measures is work on covering theorems from geometric measure theory with applications to the Henstock--Kurzweil integral of both scalar and vector valued functions. Loeb has given a stronger formulations of both the Besicovitch and the Morse Covering Theorems, and together with coauthors, he has simplified the proofs. The simple proof for Besicovitch's theorem now shows that for any norm, the best associated constant (in terms of all known proofs) is a packing constant bounded above, for a d-dimensional space, by 5^{d}.

Loeb has extending his work with Yeneng Sun on the application of Loeb measure spaces including applications in mathematical economics and game theory to obtain results that only hold when such "rich" measure spaces are used. With the appropriate map, results obtained using these measure spaces as prototype can also be shown to be valid for more general "nowhere countably generated" measure spaces.

Some Papers:

A reduction technique for limit theorems in analysis and probability theory (with J. Bliedtner), Arkiv für Matematik, 30 (1992), #1, 25-43. [reduction.pdf]

An optimization of the Besicovitch covering, Proc. Amer. Math. Soc., 118(1993), #3, 715-716. [covering.pdf]

On the best constant for the Besicovitch covering theorem (with Z. Füredi), Proc. Amer. Math. Soc., 121(1994), #4, 1063-1073. [constant.pdf]

Best filters for the general Fatou boundary limit theorem (with J. Bliedtner), Proc. Amer. Math. Soc., 123(1995), #2, 459-463. [filters.pdf]

Nonstandard integration theory in topological vector lattices (with H. Osswald), Monatshefte fur Math. 124(1997), 53--82. [lattices.pdf]

Covering theorems and Lebesgue integration (with E. Talvila), Scientiae Mathematicae Japonicae 53(2001), 209--221. [integration.pdf]

Sturdy harmonic functions and their integral representations, (with J. Bliedtner), Positivity 7(2003), 355--387. [sturdy.pdf]

Lusin's theorem and Bochner integration (with E. Talvila), Scientiae Mathematicae Japonicae 60(2004), 113--120. [BochnerJAMS.pdf]

Uncorrelatedness and orthogonality for vector-valued processes, (with H. Osswald, Y. Sun, and Z. Zhang), Tran. .Amer. Math. Soc. 356(2004), 3209--3225. [singt.pdf]

Purification of measure-valued maps (with Y. Sun), Doob Memorial Volume of the Illinois Journal of Mathematics, 50(2006), 747-762. [Loeb-Sun.pdf]

A local maximal function simplifying measure differentiation (with J. Bliedtner), MAA Monthly, 114(2007), 532--536. [LocalMax.pdf]

A general Fatou lemma (with Y. Sun). Advances in Mathematics, 213(2007), 741--762. [Fatou.pdf]

Representing measures in potential theory and an ideal boundary, Illinois Journal of Mathematics, 54(2010), 1451-1461. [BurkholderArticle.pdf]

End Compactifications and General Compactifications, (with M. Insall and M. Marciniak), Journal of Logic and Analysis 6:7(2014) 1--16.