Littlewood's three principles of real analysis


Littlewood's three principles of real analysis

Littlewood's three principles of real analysis are heuristics of J. E. Littlewood to help teach the essentials of measure theory in mathematical analysis.

The principles

Littlewood stated the principles in his 1944 "Lectures on the Theory of Functions" [cite book
last=Littlewood
first=J. E.
authorlink=John Edensor Littlewood
title=Lectures on the Theory of Functions
year=1944
publisher=Oxford University Press
oclc=297140
] Rp|26as:

The first principle is based on the fact that the inner measure and outer measure are equal for measurable sets, the second is based on Lusin's theorem, and the third is based on Egorov's theorem.

Example

Littlewood's three principles are quoted in several real analysis texts, for example Royden,cite book
last = Royden
first = H. L.
title = Real Analysis
publisher = Macmillan
location = New York
edition=3rd edition
year = 1988
isbn = 9780024041517
] Rp|72Bressoud, [] Rp|191and Stein & Shakarchi. [cite book
last = Stein
first = Elias
authorlink = Elias M. Stein
coauthors = Rami Shakarchi
title = Real Analysis: Measure Theory, Integration, and Hilbert Spaces
url = http://press.princeton.edu/chapters/s8008.pdf
format = PDF
accessdate = 2008-07-03
publisher = Princeton University Press
location = Princeton
year = 2005
isbn = 9780691113869
] Rp|33Unfortunately most authors quote the principles to point out which theorems are the most important, not as a heuristic.

RoydenRp|84 gives the bounded convergence theorem as an application of the third principle. The theorem states that if a uniformly bounded sequence of functions converges pointwise, then their integrals converge to the integral of the limit function. If the convergence were uniform this would be a trivial result, and Littlewood's third principle tells us that the convergence is almost uniform, that is, uniform outside of a set of arbitrarily small measure. Because the sequence is bounded, the contribution to the integrals of the small set can be made arbitrarily small, and the integrals on the remainder converge because the functions are uniformly convergent there.

Notes


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • John Edensor Littlewood — Infobox Scientist name = John Edensor Littlewood imagesize = 150px caption = birth date = birth date|1885|6|9|mf=y birth place = Rochester, Kent, England death date = death date and age|1977|9|6|1885|6|9|mf=y death place = Cambridge, England… …   Wikipedia

  • List of mathematics articles (L) — NOTOC L L (complexity) L BFGS L² cohomology L function L game L notation L system L theory L Analyse des Infiniment Petits pour l Intelligence des Lignes Courbes L Hôpital s rule L(R) La Géométrie Labeled graph Labelled enumeration theorem Lack… …   Wikipedia

  • theatre — /thee euh teuhr, theeeu /, n. theater. * * * I Building or space in which performances are given before an audience. It contains an auditorium and stage. In ancient Greece, where Western theatre began (5th century BC), theatres were constructed… …   Universalium

  • Chaos theory — This article is about chaos theory in Mathematics. For other uses of Chaos theory, see Chaos Theory (disambiguation). For other uses of Chaos, see Chaos (disambiguation). A plot of the Lorenz attractor for values r = 28, σ = 10, b = 8/3 …   Wikipedia

  • Queen's Medal — Die Royal Medal, auch The Queen s Medal genannt, ist eine von der Royal Society verliehene Auszeichnung für Wissenschaftler, die innerhalb des Commonwealth of Nations besonders wichtige Beiträge zur Weiterentwicklung der Wissenschaften geleistet… …   Deutsch Wikipedia

  • Ronald Fisher — R. A. Fisher Born 17 February 1890(1890 02 17) East Finchley, London …   Wikipedia

  • Royal Medal — Die Royal Medal, auch The Queen s Medal genannt, ist eine von der Royal Society verliehene Auszeichnung für Wissenschaftler, die innerhalb des Commonwealth of Nations besonders wichtige Beiträge zur Weiterentwicklung der Wissenschaften geleistet… …   Deutsch Wikipedia

  • theatre, Western — ▪ art Introduction       history of the Western theatre from its origins in pre Classical antiquity to the present.       For a discussion of drama as a literary form, see dramatic literature and the articles on individual national literatures.… …   Universalium

  • Number theory — A Lehmer sieve an analog computer once used for finding primes and solving simple diophantine equations. Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers (the… …   Wikipedia

  • Theatre — For other uses, see Theatre (disambiguation). Sarah Bernhardt as Hamlet, in 1899 Theatre (or in American English theater[1]) is a collaborative form of fine art that uses live performers to present the experience …   Wikipedia


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.