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