real-analysis-class23-24 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 23- Class 24 Sec.2.11 (Proper) Riemann integral f bdd function on [a ,b ] For any partition  : a x0 x1  ... xn b,  max{xi xi -1 : 1 i n} S upper Darboux sum s lower Darboux sum  T Riemann sum a x1 ... xn1 b  x x 0 n b (1) Darboux integral: a f (x )dx  lim S  lim s   0  0 b (2) Riemann integral: a f (x )dx  limT  0 Note: From advanced calculus, (1) (2) the same. (3) Lebesgue integral: [ , ]f (x )dx a b Thm 1.f bdd on [a ,b ] Then Riemann integrable iff conti. a.e. on [ , ] f f a b Ex. 1. monotone on [ , ] f a b disconti. at most countable (Ex.2.11.2) f Riemann integrable Ex. 2. of bdd variation on [ , ] f a b f Riemann integrable Thm 2.f bdd on [a ,b ] b Thenf Riemann integrable f Lebesgue integrable  f (x )dx [ , ]f (x )dx a a b Note 1. not true on infinite interval sin x Ex. (Ex.2.11.3) f (x ) on (1,) x Thenf Riemann integrable, but not Lebesgue integrable ( f