Carathéodory's theorem (conformal mapping): Difference between revisions
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{{about|the theorem of real analysis|the separation thereom in descriptive set theory|Lusin's separation theorem}} | |||
In the [[mathematics|mathematical]] field of [[real analysis]], '''Lusin's theorem''' (or '''Luzin's theorem''', named for [[Nikolai Luzin]]) states that every [[measurable function]] is a [[continuous function]] on nearly all its domain. In the [[Littlewood's second principle|informal formulation]] of [[J. E. Littlewood]], "every measurable function is nearly continuous". | |||
==Classical statement== | |||
For an interval [''a'', ''b''], let | |||
:<math>f:[a,b]\rightarrow \mathbb{C}</math> | |||
be a measurable function. Then, for every ''ε'' > 0, there exists a compact ''E'' ⊂ [''a'', ''b''] such that ''f'' restricted to ''E'' is continuous and | |||
:<math>\mu ( E ) > b - a - \varepsilon.\,</math> | |||
Note that ''E'' inherits the [[subspace topology]] from [''a'', ''b'']; continuity of ''f'' restricted to ''E'' is defined using this topology. | |||
==General form== | |||
Let <math>(X,\Sigma,\mu)</math> be a [[Radon measure]] space and ''Y'' be a [[second-countable]] topological space, let | |||
:<math>f: X \rightarrow Y</math> | |||
be a measurable function. Given ε > 0, for every <math>A\in\Sigma</math> of finite measure there is a closed set ''E'' with ''µ(A \ E) < ε'' such that ''f'' restricted to ''E'' is continuous. If ''A'' is [[locally compact]], we can choose ''E'' to be compact and even find a continuous function <math>f_\varepsilon: X \rightarrow Y</math> with compact support that coincides with ''f'' on ''E''. | |||
Informally, measurable functions into spaces with countable base can be approximated by continuous functions on arbitrarily large portion of their domain. | |||
==A proof of Lusin's theorem== | |||
Since ''f'' is measurable, there exists a sequence of [[step function]]s, ''f''<sub>''n''</sub> converging to f pointwise [[almost everywhere]]. Each ''f''<sub>''n''</sub> is bounded on a set of finite measure, hence [[integrable]]. By [[Egorov's theorem]], may take a closed set E, such that the measure of ''A \ E'' is arbitrarily small, and such that ''f''<sub>''n''</sub> converges to ''f'' [[Uniform convergence|''uniformly'']]. Thus ''f'' is in [[Lp space#Dense subspaces|L<sup>1</sup>]](''A''). Since continuous functions are [[dense set|dense]] in [[Lp space#Dense subspaces|L<sup>1</sup>]], we may approximate ''f'' with a [[continuous function]] defined on A. | |||
== References == | |||
* N. Lusin. Sur les propriétés des fonctions mesurables, ''Comptes Rendus Acad. Sci. Paris'' 154 (1912), 1688-1690. | |||
* G. Folland. Real Analysis: Modern Techniques and Their Applications, 2nd ed. Chapter 2 | |||
* W. Zygmunt. Scorza-Dragoni property (in Polish), UMCS, Lublin, 1990 | |||
[[Category:Theorems in real analysis]] | |||
[[Category:Theorems in measure theory]] | |||
[[Category:Articles containing proofs]] | |||
Latest revision as of 18:05, 28 January 2014
29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church.
In the mathematical field of real analysis, Lusin's theorem (or Luzin's theorem, named for Nikolai Luzin) states that every measurable function is a continuous function on nearly all its domain. In the informal formulation of J. E. Littlewood, "every measurable function is nearly continuous".
Classical statement
For an interval [a, b], let
![{\displaystyle f:[a,b]\rightarrow \mathbb {C} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/808e07bf306bb74c974003e5e32cd5dc306fd093)
be a measurable function. Then, for every ε > 0, there exists a compact E ⊂ [a, b] such that f restricted to E is continuous and
Note that E inherits the subspace topology from [a, b]; continuity of f restricted to E is defined using this topology.
General form
Let be a Radon measure space and Y be a second-countable topological space, let
be a measurable function. Given ε > 0, for every of finite measure there is a closed set E with µ(A \ E) < ε such that f restricted to E is continuous. If A is locally compact, we can choose E to be compact and even find a continuous function with compact support that coincides with f on E.
Informally, measurable functions into spaces with countable base can be approximated by continuous functions on arbitrarily large portion of their domain.
A proof of Lusin's theorem
Since f is measurable, there exists a sequence of step functions, fn converging to f pointwise almost everywhere. Each fn is bounded on a set of finite measure, hence integrable. By Egorov's theorem, may take a closed set E, such that the measure of A \ E is arbitrarily small, and such that fn converges to f uniformly. Thus f is in L1(A). Since continuous functions are dense in L1, we may approximate f with a continuous function defined on A.
References
- N. Lusin. Sur les propriétés des fonctions mesurables, Comptes Rendus Acad. Sci. Paris 154 (1912), 1688-1690.
- G. Folland. Real Analysis: Modern Techniques and Their Applications, 2nd ed. Chapter 2
- W. Zygmunt. Scorza-Dragoni property (in Polish), UMCS, Lublin, 1990