View source for Littlewood's three principles of real analysis

{{Short description|Heuristics in measure theory}}
'''Littlewood's three principles of [[Real analysis|real analysis]]''' are heuristics of J. E. Littlewood to help teach the essentials of measure theory in [[Mathematical analysis|mathematical analysis]].

==The principles==

Littlewood stated the principles in his 1944 ''Lectures on the Theory of Functions''
<ref>
{{cite book
|last=Littlewood
|first=J. E.
|title=Lectures on the Theory of Functions
|url=https://archive.org/details/in.ernet.dli.2015.89999
|year=1944
|publisher=Oxford University Press
|oclc=297140
|page=[https://archive.org/details/in.ernet.dli.2015.89999/page/n29 26]
}}
</ref>
as:
{{quote|There are three principles, roughly expressible in the following terms: Every (measurable) set is nearly a finite sum of intervals; every function (of class ''L''<sup>p</sup>) is nearly [[Continuous function|continuous]]; every [[Limit of a sequence|convergent]] sequence of functions is nearly uniformly convergent.}}

The first principle is based on the fact that the [[Inner measure|inner measure]] and [[Outer measure|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,<ref name="Royden">
{{cite book 
| last = Royden 
| first = H. L.
| title = Real Analysis 
| url = https://archive.org/details/realanalysis00royd_768
| url-access = limited
| publisher = Macmillan 
| location = New York 
| edition=3rd
| year = 1988 
| isbn = 978-0-02-404151-7
| page=[https://archive.org/details/realanalysis00royd_768/page/n84 72] }}
</ref>
Bressoud,<ref>
{{cite book | last = Bressoud | first = David | title = A Radical Approach to Lebesgue's Theory of Integration | url = https://archive.org/details/radicalapproacht00bres_468 | url-access = limited | publisher = Cambridge University Press | location = Cambridge | year = 2008 | isbn = 978-0-521-88474-7 | page=[https://archive.org/details/radicalapproacht00bres_468/page/n207 191] }}
</ref>
and Stein & Shakarchi.<ref>
{{cite book 
| last = Stein 
| first = Elias 
|author2=Rami Shakarchi
 | title = Real Analysis: Measure Theory, Integration, and Hilbert Spaces
| url = http://press.princeton.edu/chapters/s8008.pdf
| access-date = 2008-07-03
| publisher = Princeton University Press 
| location = Princeton 
| year = 2005 
| isbn = 978-0-691-11386-9
| page=33 }}
</ref>

Royden<ref>Royden (1988), p. 84</ref> 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 on a set of finite measure 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==
<references />

[[Category:Real analysis]]
[[Category:Measure theory]]
[[Category:Mathematical principles]]

{{Sourceattribution|Littlewood's three principles of real analysis}}