- Littlewood's three principles of real analysis
Littlewood's three principles of
real analysisare heuristicsof J. E. Littlewoodto help teach the essentials of measure theoryin mathematical analysis.
Littlewood stated the principles in his 1944 "Lectures on the Theory of Functions" [cite book
authorlink=John Edensor Littlewood
title=Lectures on the Theory of Functions
publisher=Oxford University Press
The first principle is based on the fact that the
inner measureand outer measureare equal for measurable sets, the second is based on Lusin's theorem, and the third is based on Egorov's theorem.
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
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.
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