Luzin theorem

Luzin's theorem in the theory of functions of a complex variable (the local principle of finite area) is a result of N.N. Luzin that reveals a connection between the boundary properties of an analytic function in the unit disc and the metric of the Riemann surface onto which it maps the disc (see [1], [2]).

Let $ V $ be any domain inside the unit disc $ D= \{ {z } : {| z | < 1 } \} $ of the complex $ z $-plane adjoining an arc $ \sigma $ of the unit circle $ \Gamma = \{ {z } : {| z | = 1 } \} $, and let

$$ w = f ( z) = \sum _ { k=0 } ^ \infty c _ {k} z ^ {k} $$

be a regular analytic function in $ D $. If the area of the Riemann surface that is the image of $ V $ under the mapping $ w = f ( z) $ is finite, then the series

$$ \sum _ { k=0 } ^ \infty c _ {k} z ^ {k} $$

converges almost-everywhere on $ \sigma $.

In connection with this theorem Luzin made a conjecture, also known as Luzin's problem. A point $ e ^ {i \theta _ {0} } \in \Gamma $ is called a Luzin point of the function $ w = f ( z) $ if $ w = f ( z) $ maps every disc touching $ \Gamma $ from the inside at $ e ^ {i \theta _ {0} } $ onto a domain of infinite area on the Riemann surface of $ w = f ( z) $. The Luzin conjecture is that there are bounded analytic functions in $ D $ such that every point of $ \Gamma $ is a Luzin point for them. The Luzin conjecture was first confirmed completely in 1955 (see [3]).

References

[1]	N.N. Luzin, "On localization of the finite area principle" Dokl. Akad. Nauk SSSR , 56 (1947) pp. 447–450 (In Russian) Zbl 0036.18101
[2]	N.N. Luzin, "Collected works" , 1 , Moscow (1953) pp. 318–330 (In Russian) MR0059845
[3]	A. Lohwater, "The boundary behaviour of analytic functions" Itogi Nauki i Tekhn. Mat. Anal. , 10 (1973) pp. 99–259 (In Russian)

Comments

The reference for the solution of Luzin's problem is [a1].

References

[a1]	A.J. Lohwater, G. Piranian, "On a conjecture of Luzin" Michigan Math. J. , 3 (1955) pp. 63–68

Luzin's theorems in descriptive set theory are, by convention, split into three parts. The first and main part is directed towards the study of effective sets (analytic, Borel, Luzin (projective) sets). Here one is concerned with the Luzin separability principles and the theorem on the existence of Luzin sets of arbitrary class (cf. Luzin set). The second part is the study of problems lying on the path to the solution of the continuum hypothesis and the problem of the cardinality of $ C {\mathcal A} $- sets (cf. $ C {\mathcal A} $- set). Here one distinguishes the Luzin–Sierpiński theorem on partitioning an interval into $ \aleph _ {1} $ Borel sets, determined by the corresponding Luzin sieve, and also Luzin's covering theorem: Let $ E $ and $ A $ be disjoint analytic sets (cf. $ {\mathcal A} $- set; Analytic set) and let

$$ E \subset X \setminus A = \ \cup _ {\alpha < \omega _ {1} } A _ \alpha $$

be a decomposition of $ X \setminus A $ into constituents; then there is an index $ \alpha _ {0} < \omega _ {1} $ such that

$$ E \subset \cup _ {\alpha < \alpha _ {0} } A _ \alpha . $$

The third part contains results obtained by the use of the axiom of choice. Here one borders on philosophical work in set theory. One distinguishes Luzin's theorem on the existence of an uncountable set of the first category (cf. Category of a set) in any perfect set, and on partitioning an interval into an uncountable number of non-measurable sets. To complete this part there is Luzin's theorem on subsets of the set of natural numbers, which reflects some properties of the remainder $ \beta \mathbf N \setminus \mathbf N $ of the Stone–Čech compactification $ \beta \mathbf N $ of the natural number series $ \mathbf N $.

References

[1]	N.N. Luzin, "Collected works" , 2 , Moscow (1958) (In Russian) MR0153522

B.A. Efimov

Comments

See Luzin sieve for the definition of constituents.

"Luzin's theorem on subsets of the set of natural numbers" states that there is a family $ \{ A _ \alpha \} _ {\alpha < \omega _ {1} } $ of infinite subsets of $ \mathbf N $ such that $ A _ \alpha \cap A _ \beta $ is finite for $ \alpha \neq \beta $ and such that for any two uncountable disjoint subsets $ E $ and $ F $ of $ \omega _ {1} $ there is no subset $ C $ of $ \mathbf N $ such that for all $ \alpha \in E $: $ A _ \alpha \setminus C $ is finite and for all $ \alpha \in F $: $ A _ \alpha \cap C $ is finite. A family like this usually called a Luzin family. See [a2].

In the West, the name "Luzin theorem" refers almost always to a result in measure theory; see Luzin criterion. It may also refer to the following result of Luzin in descriptive set theory: If $ P $ is a Polish space, $ Q $ a separable metrizable space and $ f $ is an injective Borel mapping from $ P $ into $ Q $, then the direct image $ f ( B) $ of any Borel subset $ B $ of $ P $ is a Borel subset of $ Q $. Luzin's covering theorem in descriptive set theory is usually called the (classical) boundness theorem in the West; it gave rise, together with the Luzin–Sierpiński theorem, Luzin sieves, etc., to the modern use of countable ordinals in this theory. See also Descriptive set theory.

References

[a1]	K. Kuratowski, "Topology" , 1 , PWN & Acad. Press (1966) (Translated from French) MR0234404 MR0217751 MR0217750 Zbl 0163.17002 Zbl 0158.40901 Zbl 0158.40802
[a2]	E.K. van Douwen, "The integers and topology" K. Kunen (ed.) J.E. Vaughan (ed.) , Handbook of Set-Theoretic Topology , North-Holland (1984) pp. 111–167 Zbl 0561.54004
[a3]	R. Engelking, "Hausdorff's gaps and limits and compactifications" , Theory of Sets and Topology (in honour of F. Hausdorff) , Deutsch. Verlag Wissenschaft. (1972) pp. 89–94 MR341406