Peter Albert Loeb (born July 3, 1937) is a mathematician at the

University of Illinois at Urbana–Champaign The University of Illinois Urbana-Champaign (U of I, Illinois, University of Illinois, or UIUC) is a public land-grant research university in Illinois in the twin cities of Champaign and Urbana. It is the flagship institution of the Univer ...

. He co-authored a basic reference text on

nonstandard analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using (ε, δ)-definitio ...

(Hurd–Loeb 1985). Reviewer Perry Smith for

MathSciNet MathSciNet is a searchable online bibliographic database created by the American Mathematical Society in 1996. It contains all of the contents of the journal ''Mathematical Reviews'' (MR) since 1940 along with an extensive author database, links ...

wrote: :This book is a welcome addition to the literature on nonstandard analysis. The notion of

Loeb measure In mathematics, a Loeb space is a type of measure space introduced by using nonstandard analysis. Construction Loeb's construction starts with a finitely additive map \nu from an internal algebra \mathcal A of sets to the nonstandard reals. Def ...

named after him has become a standard tool in the field. In 2012 he became a fellow of the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

.List of Fellows of the American Mathematical Society
retrieved 2013-02-02.

Notes

Publications

*Hurd, Albert E.; Loeb, Peter A. ''An introduction to nonstandard real analysis''. Pure and Applied Mathematics, 118. Academic Press, Inc., Orlando, FL, 1985. *Loeb, Peter A. "Conversion from nonstandard to standard measure spaces and applications in probability theory". ''Trans. Amer. Math. Soc.'' 211 (1975), 113–122. *Loeb, Peter A. "A new proof of the Tychonoff theorem". '' American Mathematical Monthly'' 72 1965 711–717. *Bliedtner, J.; Loeb, P. "A reduction technique for limit theorems in analysis and probability theory". Ark. Mat. 30 (1992), no. 1, 25–43. *Loeb, Peter A. "Weak limits of measures and the standard part map". ''Proceedings of the American Mathematical Society'' 77 (1979), no. 1, 128–135. *Füredi, Zoltán; Loeb, Peter A. "On the best constant for the Besicovitch covering theorem". ''Proc. Amer. Math. Soc.'' 121 (1994), no. 4, 1063–1073. *Loeb, Peter A. "A nonstandard functional approach to

Fubini's theorem In mathematical analysis Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if th ...

". ''Proc. Amer. Math.'' Soc. 93 (1985), no. 2, 343–346. *Loeb, Peter; Sun, Yeneng: "Purification of measure-valued maps". ''Illinois Journal of Mathematics'' 50 (2006), no. 1-4, 747–762. *Loeb, Peter A.; Osswald, Horst "Nonstandard integration theory in topological vector lattices". Monatsch. Math. 124 (1997), no. 1, 53–82. *Loeb, Peter A. "An axiomatic treatment of pairs of

elliptic differential equation Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form :Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + Fu +G= 0,\, wher ...

s". ''Annales de l'Institut Fourier'' (Grenoble) 16 1966 fasc. 2, 167–208.

External links

Peter Loeb
University of Illinois at Urbana-Champaign {{DEFAULTSORT:Loeb, Peter A. Mathematical logicians Living people University of Illinois Urbana-Champaign faculty Fellows of the American Mathematical Society 1937 births People from Berkeley, California

See also

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External links