实变函数第一章作业答案(周民强版).pdf

? ?1 ? 1? 1.1?8? 1.?A, B, E?X?B = ?E A?c ?Ec A? Bc= E. ? B = ?E A?c ?Ec A? B = ?Ec Ac? ?Ec A? B = Ec ?Ac A? B = Ec Bc= E. 2.?A1 An , B1 Bn ,? ? n=1 An ? ? n=1 Bn ? = n=1 ?A n Bn ?. ?x n=1 ?A n Bn ?, ?n0 1,?x An0?x Bn0.?x n=1 An,? x n=1 Bn.?x ? n=1 An ? ? n=1 Bn ? . ?x ? n=1 An ? ? n=1 Bn ? ,?x n=1 An,?x n=1 Bn.?n1 1,?x An1, ?n2 1,?x Bn2.?n0= max ?n 1, n2 ?, ? ?A n ? ? ?B n ? ? x An0?x Bn0.?x n=1 ?A n Bn ?. 3.?A1 An , B1 Bn ,? ? n=1 An ? ? n=1 Bn ? = n=1 ?A n Bn ?. ?Ac 1 Ac2 Acn , Bc1 Bc2 Bcn , ?2? ? n=1 Ac n ? ? n=1 Bc n ? = n=1 ?Ac n B c n ?, ? ? n=1 An ?c ? n=1 Bn ?c = n=1 ?A n Bn ?c, ? ? n=1 An ? ? n=1 Bn ? = ? n=1 ?A n Bn ?c?c = n=1 ?A n Bn ?. 1.2?11? 1.?An= n m n ? ? ?m Z o , Bn= ? 1 n, 1 + 1 n ? , n = 1, 2, ,? (i) lim n An= Q,(ii) lim n An= Z,(iii) lim n Bn= (0, 1. ?(i)?x lim n An= j=1 n=j An,? j 1, n0 j,?x An0,? m0 Z,? x = m0 n0 Q.?lim n An Q. ?x Q,? n1 N,? m1 Z,?x = m1 n1 .?j 1,?n0= j n1 ?m0= j m1,?x = m1 n1 = j m1 j n1 = m0 n0 An0,?x lim n An.?Q lim n An. ? n 1,?An Q,?lim n An Q.?r = p q Q,?r Akq( k 1), ?r? ?A n ? ?r lim n An.?Q lim n An.? (ii) x lim n An= j=1 n=j An,? j0 1, n j0,?x An,?x Aj0?x Aj0+1.? m1, m2 Z,?x = m1 j0 = m2 j0+ 1. ?x = m2 m1 Z.?lim n An Z. ?x Z,? m1 Z,?x = m1.?j0= 1,?n j0= 1,? m0= m1n,?x = m1= m1 n n = m0 n An,?x lim n An.?Z lim n An.? n 1, ?Z An,?Z lim n An.? (iii)?lim n Bn (0, 1 lim n Bn. ?x lim n Bn= j=1 n=j Bn,? j 1, nj j,?x Bnj= ? 1 nj, 1 + 1 nj ? .? 1 nj x 1 + 1 nj. ?j ,?0 1 n1 x 1,?x (0, 1.?lim n Bn (0, 1. ?x (0, 1,?j0 1,? 1 j0 x,? n j0,? 1 n 1 j0 x 1 1 + 1 n, ?x Bn.?x lim n Bn.?(0, 1 lim n Bn. ?Bn= ? 1 n, 1 + 1 n ? = ? 1 n, 1 ? h 1, 1 + 1 n ? = ? 1 n, 2 ? ? 0, 1 + 1 n ? .? Cn= ?1 n, 1 ? ,Dn= h 1, 1 + 1 n ? ,En= ?1 n, 2 ? ,Fn= ? 0, 1 + 1 n ? , ?Cn, En?Dn, Fn? lim n Cn= ?0, 1?, lim n Dn= ?1?, lim n En= ?0, 2?, lim n Fn= ?0, 1?. ?63?3? lim n Bn= lim n Cn lim n Dn= (0, 1,lim n Bn= lim n En lim n Fn= (0, 1. 2 2.? ?f k(x) ? k1 ?f(x)?R1?lim n fn(x) = f(x), x R1,? ?t R1,? n x R1 ? ? ?f(x) t o = k=1 m=1 n=m n x R1 ? ? ?f n(x) t + 1 k o . ?A = n x R1 ? ? ?f(x) t o ?An, k= n x R1 ? ? ?f n(x) t+ 1 k o ,?A = k=1 m=1 n=m An, k. ?x0 A,?f(x0) t.?lim n fn(x0) = f(x0),? k 1, m 1,? n m, ?fn(x0) f(x0)+ 1 k t+ 1 k, ?x0 An,k.?x0 k=1 m=1 n=m An, k.?A k=1 m=1 n=m An, k. ?x0 k=1 m=1 n=m An,k,? k 1, m 1,? n m,?fn(x0) t + 1 k. ? lim n fn(x0) = f(x0),?k, n ,?f(x0) t,?x0 A.? k=1 m=1 n=m An, k A. 3.?lim n an= a,? ?a? = k=1 N=1 n=N ? an 1 k , an+ 1 k ? . ?An,k= ? an 1 k, an + 1 k ? .?lim n an= a,?k 1,?N 1,? ?n N,? ? ?a n a ? ?1 k, ?an 1 k a an+ 1 k, ?a An,k,?a k=1 N=1 n=N An,k. ?x k=1 N=1 n=N An,k,?k 1,?N 1,?n N,? x An,k,?an 1 k 0,?B(x, ) E 0,?B(x, ) Ec= x / Ec?x / (Ec) x / Ec x ?Ec?c; x E 0, y E?z Ec,?y B(x, ), z B(x, ) x E, x / E x E E. 4.?(i) G Rn? G G = ;(ii) F Rn? F F. ?(i)?G?G?GG = . ?G G = .?G?x G,?x / G,? ? 0,?y Gc,?y / B(x, ),?B(x, ) G. (ii)?2,?F = F F F = F. ?F F.?x F,? 0,? ?B(x, )x? F 6= .?x F, ?x / F,?x F F.?x F.?F F,? F? ?F = Fc,? F? Fc? Fc Fc= Fc F = F F. 9.?Rn? ?G ? ?E? ?G ? ?E? ?E = (0, 1), Gk= (1 k, 1), ?E k=1 Gk,?E = 0, 1 * (0, 1 = k=1 Gk. ?E?R2?E = n (x, y) R2 ? ? ?x R, y 0 o .? ? = n B(x, r) E ? ? ?x Q2 E, r Q+ o . ?E? = n B(x, r) ? ? ?B(x, r) o ,?x? ?E = n (x, y) R2 ? ? ?x R, y 0 o ?R2?E. 5 13.?F R1?f C(F),? n x F ? ? ?f(x) = 0 o ? ?E = n x F ? ? ?f(x) = 0 o .? ?x n ? E?lim n xn= x.?x E.?xn F? f(xn) = 0,?F?x F.?f C(F),?f(x) = lim n f(xn) = 0,?x E. 1.8?62? 1.?E Rn?x / E,?y E,?d(x, y) = d(x, E),?E ? ?(?).?E* E,?x0 E,?x0/ E.?d(x0, E) = 0,? ?y E,?d(x0, y) 6= 0.?d(x0, y) 6= d(x0, E),? ?Ec?x / E,?y E,?d(x, y) = d(x, E). ? = d(x,y) 2 0,?B(x, ) E = ,?B(x, ) Ec.?x?Ec?x? ?Ec? 2? 2.1?63? 3.? ?A n ? ? ?B n ?, ? (i) lim n ?A n Bn ? = lim n An lim n Bn;(ii) lim n ?A n Bn ? = lim n An lim n Bn; ?lim n ?A n Bn ? lim n An lim n Bn?lim n ?A n Bn ? lim n An lim n Bn.? ?(i)? n n=j An o j1 ? n n=j Bn o j1 ?8?2? lim n ?A n Bn ? = j=1 n=j ?A n Bn ? = j=1 ? n=j An ? ? n=j Bn ? = ? j=1 n=j An ? ? j=1 n=j Bn ? = lim n An lim n Bn. (ii)? n n=j An o j1 ? n n=j Bn o j1 ?8?3? lim n ?A n Bn ? = j=1 n=j ?A n Bn ? = j=1 ? n=j An ? ? n=j Bn ? = ? j=1 n=j An ? ? j=1 n=j Bn ? = lim n An lim n Bn. 6 22.?G1?G2?Rn?G1 G2= . ?G1 G2= ,?G2 Gc 1. ?G1?G2 Gc 1= Gc1. ?G1 G2= . 23.?G Rn.?E Rn,?G E G E,?G? ?E = Gc,?G Gc G Gc= ,?Gc Gc.?Gc?G? 27.? ?F ? ?Rn?F1, F2, , Fm,? m i=1 Fi6= , ? F6= . ? F= ,? Fc = ? F ?c = Rn.? ?F ? ?F1,? ?F c ? ?F1?Borel?m 1,?F1 m k=2 Fc k = ? m k=2 Fk ?c . ? m k=1 Fk= F1 ? m k=2 Fk