real-analysis-class33 实变函数论应用数学讲义.pdf

實變函數論─應用數學系 吳培元老師 Class 33 Fubinis Thm h : X Y  integrable Then (1) hx integrable on Y for almost all x ; (2) hy integrable on X for almost all y ; (3)f (x) h d  integrable on X , g (y ) hy du integrable on Y ; x (4) hd (u ) hdud hdud ( )          Pf: Let h h h May assume h  0, integrable Tonellis  hd (u ) gd  f du (i.e., (4))    f ,g integrable (i.e., (3)) f ,g finite a.e.  h d  x i.e., hx integrable for almost all x (i.e., (1)) Note: Difficult to obtain u  diffi. to determine the integrability of h Combination of Tonelli Fubini: Cor.1. h : X Y [ , ] measurable  h dud or  h ddu  Then h integrable on X Y ( Preceding Thm applicable) Pf: Tonellis for h  hd (u ) h dud h ddu     i.e., h integrable h integrable Special cases: Let X Y 1,2,3,...   X   2 u  counting mea.  X Y       u  E   E i j Then 2 , counting measure ( , ( , ) )    ( , ) i j E (1)h : X Y [0, ], i.e., amn  0 (m, n)  am