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Curso: Análisis Real II Clase 3: Aproximación De Funciones CONTENIDO REFERENCIAS SUGERIDAS CURSO:ANÁLISIS REAL II CLASE 3: APROXIMACIÓN DE FUNCIONES MEDIBLES POR FUNCIONES CONTINUAS Yamilet Quintana1 1Universidad Simón Bolívar Programa Interuniversitario de Doctorado en Matemáticas Junio, 2020 CONTENIDO REFERENCIAS SUGERIDAS 1 CONTENIDO Aproximación de funciones medibles por funciones continuas p Densidad de Cc(X) en L (X; µ), 1 ≤ p < 1 El teorema de Luzin 2 REFERENCIAS SUGERIDAS CONTENIDO REFERENCIAS SUGERIDAS 1 CONTENIDO Aproximación de funciones medibles por funciones continuas p Densidad de Cc(X) en L (X; µ), 1 ≤ p < 1 El teorema de Luzin 2 REFERENCIAS SUGERIDAS CONTENIDO REFERENCIAS SUGERIDAS APROXIMACIÓN DE FUNCIONES MEDIBLES POR FUNCIONES CONTINUAS Aproximación de funciones medibles por funciones continuas Para finalizar el Tema 1, en esta lección nos ocuparemos de dos resultados sobre aproximaxión de funciones: El teorema de densi- dad de las funciones continuas con soporte compacto en Lp(X; µ), 1 ≤ p < 1, y el teorema de Luzin sobre aproximación de funciones medibles por funciones continuas. Nuestro objetivo será enunciar y demostrar ambos resultados. CONTENIDO REFERENCIAS SUGERIDAS p DENSIDAD DE Cc (X) EN L (X; µ), 1 ≤ p < 1 p Densidad de Cc(X) en L (X; µ), 1 ≤ p < 1 TEOREMA 1-C3 Sean (X; τ) Hausdorff localmente compacto, B(X) su σ-álgebra de Borel, µ : B(X) ! [0; 1] regular, y 1 ≤ p < 1. Entonces Cc (X) es denso en Lp(X; µ). Demostración Seguiremos los enfoques de las referencias [2,10]. p Paso 1: Verifiquemos que Cc (X) es un subconjunto de L (X; µ). En efecto, si f 2 Cc (X) entonces f es acotada y Z p p p kf kLp(X,µ) = jf (x)j dµ(x) ≤ µ(supp(f )) kf k1 : X CONTENIDO REFERENCIAS SUGERIDAS p DENSIDAD DE Cc (X) EN L (X; µ), 1 ≤ p < 1 Demostración (cont.) Paso 2: Sea S el conjunto de las funciones simples que se anulan fuera de conjuntos de medida cero. Un razonamiento similar al usado en la demostración del Theorem 4.7 (ver Real Analysis: Chapter 4, Prof. W. Urbina), nos permite mostrar que S es denso en Lp(X; µ). Paso 3: Densidad de Cc (X) en S: Sean E 2 B(X) tal que µ(E) < 1 y " > 0, " p convenientemente tomaremos δ = 2 . Usando que µ es regular, encontramos K compacto y U abierto tales que K ⊂ E ⊂ U, µ(K ) > µ(E) − δ y µ(U) < µ(E) + δ. Luego, aplicando el LEMA DE URYSOHN, encontramos g 2 Cc (X) con 0 ≤ g ≤ 1 tal que χK ≤ g y supp(g) ⊂ U. Entonces p p kg − χE kLp(X,µ) = k(g − χK ) + (χK − χE )kLp(X,µ) p p ≤ kg − χK kLp(X,µ) + kχK − χE kLp(X,µ) 1 1 1 = (µ(U n K )) p + (µ(E n K )) p < 2(δ) p = ": CONTENIDO REFERENCIAS SUGERIDAS p DENSIDAD DE Cc (X) EN L (X; µ), 1 ≤ p < 1 Demostración (cont.) p p Paso 4: Densidad de Cc (X) en L (X; µ): Finalmente, sea h 2 L (X; µ) por el Paso 2 existe s 2 S tal que " kh − skp < ; Lp(X,µ) 2 ahora por el Paso 3 para la función simple s podemos encontrar g 2 Cc (X) tal que " ks − gkp < ; Lp(X,µ) 2 por lo tanto, p p kh − gkLp(X,µ) = k(h − s) + (s − g)kLp(X,µ) " " ≤ kh − skp + ks − gkp < + = ": Lp(X,µ) Lp(X,µ) 2 2 CONTENIDO REFERENCIAS SUGERIDAS EL TEOREMA DE LUZIN El teorema de Luzin TEOREMA 2-C3: TEOREMA DE LUZIN Consideremos (X; τ) un espacio Hausdorff localmente compacto, B(X) su σ-álgebra de Borel, µ : B(X) ! [0; 1] una medida de Borel regular interior sobre X, una función medible Borel f : X ! R y un subconjunto E 2 B(X) tal que µ(E) < 1. Para cualquier " > 0 existe un compacto K ⊂ E tal que µ(E n K ) < " y f jK es continua. Demostración Seguiremos los enfoques de las referencias [2,10]. Sean E 2 B(X) tal que µ(E) < 1 y " > 0. Consideremos fIngn2N una sucesión de todos los intervalos abiertos con extremos racionales. Para cada n 2 N la regularidad interior de µ implica la existencia de conjuntos −1 −1 compactos Cn; Dn ⊂ E tales que Cn ⊂ f (In) y Dn ⊂ f (R n In), y " µ(E n (C [ D )) < : n n 2n CONTENIDO REFERENCIAS SUGERIDAS EL TEOREMA DE LUZIN Demostración (cont.) Veremos que el conjunto \ K := (Cn [ Dn) n2N satisface los requerimientos del Teorema. Es claro que K es compacto, y también que X µ(E n K ) ≤ µ(E n (Cn [ Dn)) < ": n2N Por otro lado, si x 2 K y V es un entorno abierto de f (x), entonces existe n0 2 N tal que x 2 In0 ⊂ V . Entonces el conjunto W = Cn0 \ K = (X n Dn0 ) \ K es un entorno abierto de x en K y f (W ) ⊂ In0 ⊂ V . Por lo tanto f jK es continua en cada x 2 K . CONTENIDO REFERENCIAS SUGERIDAS EL TEOREMA DE LUZIN Observaciones: (1) El enunciado original del teorema demostrado por Luzin es el siguiente: Sea f1(x); f2(x);:::; fn(x);::: una sucesión de funciones medibles que suponemos convergente y tiende a una función-límite f (x) para x en el intervalo 0 ≤ x ≤ 1, excepto quizás en los puntos de un conjunto de medida cero. Según un importante teorema del Sr. Egoroff (Comptes rendus, 30 de enero de 1911), existe un conjunto P perfecto y no denso con las siguientes propiedades: 1) La sucesión f1; f2;:::; fn;::: converge uniformemente a f en este conjunto P. 2) La medida de P es mayor que 1 − ",(" > 0 tan pequeño como se quiera). Suponiendo que fn(x) es un polinomio (n = 1; 2; 3; ldots) obtendremos el siguiente resultado: si f (x) es una función definida en el intervalo 0 leqx ≤ 1 de clase 1 (según el Sr. Baire), entonces en este intervalo existe un subconjunto perfecto y no denso P con las propiedades: 1) La función f (x) es continua en P ( f jP es continua). 2) La medida de P es mayor que 1 − ",(" > 0 tan pequeño como se quiera). La medida a la que hace referencia Luzin, es por supuesto la medida de Lebesgue en R, al igual que la noción de funciones medibles es en el sentido Lebesgue. CONTENIDO REFERENCIAS SUGERIDAS EL TEOREMA DE LUZIN Observaciones: (2) El enunciado usual del Teorema de Luzin (para la medida de Lebesgue) en muchos libros de texto es el siguiente (ver Ejercicio 3.): Sea f :[a; b] ! R una función medible (Lebesgue). Entonces para cualquier " > 0 existe un conjunto compacto K ⊆ [a; b] tal que f jK es continua y µ(K ) > b − a − ". (3) La versión del Teorema 2-C3 para funciones f : X ! C fue estudiada en Theorem 4.8 (ver Real Analysis: Chapter 4, Prof. W. Urbina). CONTENIDO REFERENCIAS SUGERIDAS EL TEOREMA DE LUZIN Nikolai Nikolaevich Luzin Nikolai Nikolaevich Luzin (este apellido algunas veces también es escrito como Lusin) (1883-1950) fue un matemático soviético conocido por su trabajo en Teoría Descriptiva de Conjuntos y aspectos del Análisis Matemático con fuertes conexiones con Topología General. En 1912 Luzin viaja de Göttingen a Paris, donde permane hasta 1914. Durante este periodo publica una serie de papers, entre los cuales aparece el siguiente: N. Lusin, Sur les propriétés des fonctions mesurables, Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences 154 (1912), 1688- 1690. Disponible en: https://gallica.bnf.fr/ark:/12148/bpt6k31070/ f1.item.r=Lusin Este paper contiene la demostración original del Teorema de Luzin, dando respuesta a la formulación informal hecha por J. E. Littlewood: “cualquier función medible es casi una función continua”. CONTENIDO REFERENCIAS SUGERIDAS EL TEOREMA DE LUZIN Nikolai Nikolaevich Luzin El contenido de estas contribuciones de Luzin durante el periodo 1912-1914 (salvo una), sería luego incorporado a su tesis doctoral The Integral and Trigonometric Series (1915) dirigida por DMITRI F. EGOROFF. En la década de los años 20 del siglo pasado, Luzin organizó un famoso seminario de investigación en Moscow State University. Entre sus estudiantes doctorales se cuentan algunos de los más famosos matemáticos soviéticos: PAVEL ALEKSANDROV, Nina Bari, Aleksandr Khinchin, ANDREY KOLMOGOROV, ALEXANDER KRONROD, Mikhail Lavrentyev, Alexey Lyapunov, Lazar Lyusternik, Piotr Nóvikov, Lev Schnirelmann y PAVEL URYSOHN. Para mayor información con conexiones históricas sobre el trabajo de Luzin, consulte: E. R. Phillips, Nicolai Nicolaevich Luzin and the Moscow school of the theory of functions, Historia Math. 5 (3) (1978), 275-305. CONTENIDO REFERENCIAS SUGERIDAS Referencias sugeridas [1] J. J. Benedetto, W. Czaja, Integration and Modern Analysis, Birkhauser,˝ 2009. [2] H. Bercovici, A. Brown, C. Pearcy, Measure and Integration, Springer International Publishing, 2016. [3] P. Cannarsa, T. D’Aprile, Introduction to Measure Theory and Functional Analysis, Springer International Publishing, 2015. [4] J. B. Conway, A Course in Abstract Analysis, AMS Graduate Studies in Mathematics, Vol 41, 2012. [5] C. S. Kubrusly, Essentials of Measure Theory, Springer International Publishing, 2015. CONTENIDO REFERENCIAS SUGERIDAS Referencias sugeridas [6] H. L. Royden, Real Analysis (3rd ed), Macmillan Publishing Co., 1988. [7] H. L. Royden, P. Fitzpatrick, Real Analysis (4th ed), Prentice Hall, 2010. [8] W. Rudin, Real and Complex Analysis (3rd ed), McGraw Hill, Inc., 1987. [9] D. A. Salamon, Measure and Integration, EMS Textbooks in Mathematics, 2016. [10] M. E. Taylor, Measure Theory and Integration, AMS Graduate Studies in Mathematics, Vol 76, 2006..
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