实变函数与泛函分析基础第三版(程其襄)+课后答案.pdf

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b R f x a b L f x a b R Z a b f x dx R Z b a f x dx f x a b R 0 0 fn mE fn Z E fn fn x dx Z E fndx mE fn 1 Z E fn x dx limmE fn lim n 1 Z E fn x dx 0 fn x 0 7 mE fn a e lim n Z E fn x 1 fn x dx 0 fn x 0 fn 0 E fn 1 fn E fn limmE fn 1 fn limmE fn 0 fn 1 fn 0 0 fn 1 fn 1 n 1 2 mE 1 E fn 1 fn 1 E fn lim n mE fn lim n mE h fn 1 fn 1 i 0 fn 0 8 f x sin 1 x x 0 x 1 f x 0 1 L 3 1 Z 1 0 f x dx Z 1 0 1 x sin 1 x dx Z1 0 1 x sin 1 x dx Z siny y dy 1 f x L 0 x E f mE f Z E f f x dx Z E f x x dx 0 mE f 0 mE f 0 mE f 0 E f 0 n 1 E f 1 n mE f 0 P n 1 mE h f 1 n i 0 f x 0a e E 11 lim n Z 0 dt 1 t n n t 1 n 1 t 0 1 1 1 t n n t 1 n 1 t 1 n 1 t n 2 t 1 n 2 1 1 t n n t 1 n 1 1 t n 1 2n t2 t 1 n 2n t2 n 1 4 t2 4 F t 1 t t 0 1 4 t2 t 1 Z 0 F x dx Z 1 0 dt t Z 1 4dt t2 6 F x 0 lim n Z 0 dt 1 t n n t 1 n Z 0 lim n 1 1 t n n t 1 n dt Z 0 dt et 1 12 1 1 x 1 x x2 x3 0 x 1 ln2 1 1 2 1 3 1 4 0 1 xn xn 1 0 5 3 Z 1 0 1 1 xdx X n 0 Z 1 0 x2n x2n 1 dx X n 0 1 2n 1 1 2n 2 1 1 2 1 3 1 4 R 1 0 1 1 xdx ln2 ln2 1 1 2 1 3 1 4 13 f x t t t0 x a b K t f x t K a x b t t0 d dt Z b a f x t dt Z b a f t x t dx hn lim n hn 0 hn 0 d dt Z b a f x t dt lim n 1 hn Z b a f x t hn f x t dx f x t hn f x t hn f t x t hn hn hn f t x t hn K 0 1 a x b t0 t hn 1 5 xp 1 x ln 1 x X n 0 xn xpln 1 x X n 0 xn pln 1 x x 0 1 xn pln 1 x 0 Z 1 0 xp 1 x ln 1 x dx X n 0 Z 1 0 xn pln xdx X n 0 1 n p 1 2 X n 1 1 n p 2 15 fn E lim n fn x f x a e E Z E fn x dx 0 a b x Z b a f x x dx 0 x x a b x x x x 3 b a 0 t x a b x t t 3 b a Z b a f x t f x dx Z b a f x x dx Z b a f x t x t dx Z b a x t x dx 3 3 3 b a b a lim t 0 Z b a f x t f x dx 0 6 17 f x fn x n 1 2 E lim n fn x f x a e E lim n Z E fn x dx Z E f x dx e E lim n Z e fn x dx Z e f x dx Z e f x dx Z e lim n fn x dx lim n Z e fn x dx Z e f x dx lim n Z e fn x dx Z e f x dx Z e f x dx lim i Z E e fni x dx lim i Z E fni x dx lim i Z e fni x dx 0 xn 0 lim n xn f xn 0 f x 0 0 x x 0 x x f x f x 0 2 x xn xn f x f xn 0 2 f x f xn 0 2 0 2 xn xni xni 2 i 1 2 Eni xni xni Eni Z 0 f x dx Z S i 1 Eni f x dx X i 1 Z Eni f x dx 0 2 X i 1 mEni X i 1 0 7 f x 0 lim x f x 0 19 f x Rp g y Rq f x g y Rp Rq f x Rp Rp Rp Rq g y Rp Rq f x g y Rp Rq f x g y 6 4 f x g y Rp Rq ZZ Rp Rq f x g x dxdy Z Rp dx Z Rq f x g y dy Z Rp f x dx Z Rq g y dy 0 Z 0 F y dy Z E f x g x dx f x g x E y 0 Ey F y R Ey f x dx F y 0 Z 0 F y dy Z 0 Z Ey f x dx dy Z 0 Z E Ey x f x dx dy Z E f x Z g x 0 dy dx Z E f x g x dx 8 1 a b f x g x a b E a b E f x g x M f x N g x x0 M f x0 0 f x0 f x0 0 E a b xn E xn x0 lim n xn x0 f x0 0 lim n f xn lim n g xn g x0 0 f x0 0 g x0 0 x0 g x x0 E M N N M M N f x g x 2 fn a b fn x f x n f x b W a fn K n 1 2 f x a b T a x0 x1 xm b m X i 1 fn xi fn xi 1 b a fn 0 0 1 x0 0 xk h n 1 k 2 i 1 k 1 2 n 1 xn 1 n X k 1 f xk f xk 1 n X k 1 x ksin 1 x k x k 1sin 1 x k 1 n 1 X k 1 1 n 1 k 2 1 n k 2 n 1 X k 1 1 n 1 k 2 1 n k 2 n 1 X k 1 1 n k 1 n k 1 n 1 X k 1 2 n k 1 2 n X k 2 1 k 1 n P k 2 1 k 1 0 f sup X k 1 f xk f xk 1 f x 0 1 0 f x x 1sin 1 x x 1cos 1 x f x x 1 x 1 0 1 1 1 1 f x 0 1 R L f x 0 1 L Z x 0 f t dt lim 0 L Z x f t dt lim 0 R Z x f t dt lim 0 t sin 1 t x x sin 1 x 0 f x f 0 f x f 0 Z x 0 f t dt f x 0 1 4 f x a b f x 0a e a b f x x1 x2 a b x2 x1 f x a b f x1 f a Z x1 a f t dt f x2 f a Z x2 a f t dt f x2 f x1 Z x2 x1 f t dt f x 0a e a b Rx2 x1 f t dt 0 f x2 f x1 f x 5 f x a b M 0 0 b a f M f x a b x a b f x f b b x f M f x M f b a b T T a x0 x1 x2 xn b 2 T v n X i 1 f xi f xi 1 f x1 f a n X i 2 f xi f xi 1 f x1 f a b x1 f 2M f b f a b a f 2M f b f a f x a b 6 fn a b f x P n 1 fn x a b f x a b n fn x a b f