实变函数与泛函分析课后习题答案

? 1.?A (B C) = (A B) (A C). ?x (A(B C).?x A,?x AB,x AC,?x (AB)(AC). ?x B C,?x A B?x A C,?x (A B) (A C),? A (B C) (A B) (A C). ?x (A B) (A C).?x A,?x A (B C).?x 6 A,? x A B?x A C,?x B?x C,?x B C,?x A (B C),? (A B) (A C) A (B C).?A (B C) = (A B) (A C). 2.? (1)A B = A (A B) = (A B) B; (2)A (B C) = (A B) (A C); (3)(A B) C = A (B C); (4)A (B C) = (A B) (A C); (5)(A B) (C D) = (A C) (B D); (6)A (A B) = A B. ?(1)A(AB) = As(AB) = A(sAsB) = (AsA)(AsB) = AB; (A B) B = (A B) sB = (A sB) (B sB) = A B; (2)(AB)(AC) = (AB)s(AC) = (AB)(sAsC) = (AB sA)(A B sC) = A (B sC) = A (B C); (3)(A B) C = (A sB) sC = A s(B C) = A (B C); (4)A(B C) = A(B sC) = As(B sC) = A(sB C) = (AsB)(AC) = (A B) (A C); (5)(A B) (C D) = (A sB) (C sD) = (A C) s(B D) = (A C) (B D); (6)A (A B) = A s(A sB) = A (sA B) = A B. 3.?(A B) C = (A C) (B C);A (B C) = (A B) (A C). ?(A B) C = (A B) sC = (A sC) (B sC) = (A C) (B C); (AB)(AC) = (AsB)(AsC) = AsB sC = As(B C) = A(B C). 4.?s( S i=1 Ai) = T i=1 sAi. ?x s( S i=1 Ai),?x S,?x 6 S i=1 Ai,?i,x 6 Ai,?x sAi,? 1 x T i=1 sAi.?x T i=1 sAi,?i,x sAi,x S,x 6 Ai,?x S,?x 6 S i=1 Ai, ?x s( S i=1 Ai).?s( S i=1 Ai) = T i=1 sAi. 5.?(1)( S A) B = S (A B); (2)( T A) B = T (A B). ?(1) S A B = ( S A) sB = S (A sB) = S (A B); (2) T A B = ( T A) sB = T (A sB) = T (A B). 6.?An?B1= A1,Bn= An( n1 S =1 A),n 1.?Bn? ? n S =1 A= n S =1 B,1 n . ?i 6= j,?i j.?Bi Ai(1 i n). Bi Bj Ai (Aj j1 n=1 An) = Ai Aj sA1 sA2 sAi sAj1= . ?Bi Ai(1 6= i 6= n)? n S i=1 Bi n S i=1 Ai. ?x n S i=1 Ai,?x A1,?x B1 n S i=1 Bi.?x 6 A1,?in?x Ain, x 6 in1 S i=1 Ai?x Ain.?x Ain in1 S i=1 Ai= Bin n S i=1 Bi.? n S i=1 Ai= n S i=1 Bi. 7.?A2n1= ?0, 1 n ? ,A2n= (0,n),n = 1,2, ,?An? ?lim n An= (0,); ?x (0,),?N,?x N?0 N? x A2n1,0 x f(x 0). (3)?x1,x2 E,?x1 x2,?f(x10) f(x1+0) f(x20) f(x2+0),? x E,?(f(x 0),f(x+ 0),?(3)?E?x? ?11? 15.?(0,1)?0,1? 3 ?(0,1)?R = r1,r2,? (0) = r1, (1) = r2, (rn) = rn+2,n = 1,2, (x) = x,x (0,1)R), ?0,1?(0,1)? 16.?A?A? ?A = x1,x2,A? e A.An= x1,x2, ,xn,An? ? e An.? e An?2n? e A = S n=1 e An,? e A?A? ? e A? 17.?0,1?c. ?0,1?A,0,1?r1,r2,? B = ( 2 2 , 2 3 , , 2 n , ) A ? ( 2 2n ) = 2 n + 1, n = 1,2, ( 2 2n + 1) = rn, n = 1,2, (x) = x,x 6 B. ?A?0,1?0,1?c,?A?c. 18.?A?A = ax1x2x3,? ?xi?c?A?c. ?xi Ai,Ai= c,i = 1,2, .?Ai?R?i.?A? E?ax1x2x3 A.(ax1x2x3) = (1(x1),2(x2),3(x3),).? ? ?(ax1x2x3) = (ax 1x 2x 3), ?i,i(xi) = i(xi).?i?xi= xi,?ax1x2x3= ax 1x 2x 3. ?(a1,a2,a3,) E,ai R,i = 1,2, ,?i? ?xi Ai,?i(xi) = ai.?ax1x2x3 A,?(ax1x2) = (1(x1),2(x2),) = (a1,a2,),?A?E?c. 19.? S n=1 An?c,?n0,?An0?c. ?E= c,? S n=1 An= E.?An c,n = 1,2, .?Pi E?R?x = (x1,x2, ,xn,) E,?Pi(x) = xi.? A i = Pi(Ai),i = 1,2, , 4 ?A i Ai c,i = 1,2, .?i,?i RA i, ? = (i,2, ,n,) E. ? 6 S n=1 An.? S n=1 An,?i,? Ai,?i= Pi() Pi(Ai) = A i, ? ? RA i ? 6 S n=1 An= E,? E?i0,? Ai0= c. 20.?0?1?T,?T?c. ?T = 1,2, | i= 0or1,i = 1,2,. ?T?E? : 1,2, 2,3,?T?E?(T)? ?A E= c,?(0,1?2? x (0,1?x = 0.12 ,?i0?1,?f(x) = 1,2,?f? (0,1?T?f(0,1)?T (0,1 = c.?A = c. 5 ? ? ?Eo?E? E? 1.?P0 E?P0?U(P,) (?P0?)? ?P0?P1?E (?P1?),?P0 Eo? ?P0?U(P,)(?P0?)?U(P,) E. ?P0 E,?P0?U(P,),?P0?U(P0) U(P,), ?P1 E U(P0) E U(P,)?P16= P,?P0?P1? P0?E. ?P0?P0?P1?E,?P0?U(P0)? ?P0?P1?E,?P0 E. ?P0 Eo,?U(P0) E. ?P0 U(P,) E,?U(P0) U(P,) E,?P0 Eo. 2.?E1?0,1?E1?R1?E 1,Eo1, E1. ?E 1= 0,1, Eo 1 = , E1= 0,1. 3.?E2= (x,y)|x2+ y2 a.?f(x)? 0,?x (,),|xx0| a,?x U(x0,)?x E,?U(x0,) E,E? ? ?xn E,?xn x0(n ).?f(xn) a,?f(x)?f(x0) = lim n f(xn) a, ?x0 E,?E? 9.? ?F?Gn= ?x|d(x,F) 1 n ?,G n ?x0 Gn,d(x0,F) 1 n, ? ?y0 F,?d(x0,y0) = 1 n. ( ?y F,d(x0,y) 1 n, ?d(x0,F) = inf yF d(x0,y) 1 n, ?d(x0,F) 0,?x U(x0,),d(x0,x) . d(x,y0) d(x0,x) + d(x0,y0) + = + 1 n = 1 n. ?d(x,F) = inf yF d(x,y) d(x,y0) 1 n, ?x Gn.?U(x0,) Gn,?Gn? ?x T n=1 Gn,?n,x Gn,d(x,F) 0,xn x0,f(xn) f(x0) + 0 ?f(xn) f(x0) 0,?c = f(x0) + ,?xn E = x|f(x) c,? x06 E(?f(x0) t0,tn t0?Ptn E,?Ptn Pt0,?Pt0 E. ?Pt06 E,?t06= 0,?tn,0 tn t0,tn t0,Ptn Pt0,Ptn E,? Pt0 E.?E 6= . 13.?P?1,? P?c. ?P?(?P),? ?1 3, 2 3 ? = (0.1,0.2), ?1 9, 2 9 ? = (0.01,0.02), ?7 9, 8 9 ? = (0.21,0.22), ?n?2n1?I(n) k ,k = 1,2, ,2n1? I(n) k = (0.a1a2an11,0.a1a2an12), ?a1,a2, ,an1?0?2.?0,1 P? ?1,?P?1,?x P,?x? ? x = a1 3 + a2 32 + + an 3n + , ?an?0?2.?A,?A P.?A 0,1,?0,1 P ?ai?1,?0,1 P?A? A P. 3 ?A?B?: : x = X n=1 an 3n X n=1 1 2n an 2 , ?an= 0?2,?A?B?1-1?A?c,?A P,? P c,? P c ,? P = c. 4 ? ? 1.?E?mE 0,?mE?c,?E? ?E1,?mE1= c. ? a = inf xE x,b = sup xE x,?E a,b.?Ex= a,x E,a x b,f(x) = mEx? a,b?x 0? | f(x + x) f(x) | =| mEx+x mEx| | m(Ex+ E) | m(x,x + x = x. ?x 0?f(x+x) f(x),?f(x)?x 0,x 0 ?f(x x) f(x),?f(x)?a,b? f(a) = mEa= m(E a) = 0 f(b) = m(E a,b) = mE. ?c,c mE,?x0 a,b?f(x0) = c.?mEx0= m(a,x0 E) = c.? E1= E a,x0 E.?mE1= c. 4.? S1,S2, ,Sn?,Ei Si,i = 1,2. ,n,? m(E1 E2 En) = mE1+ mE2+ + mEn. ?S1,S2, ,Sn?2?3?1,? ?T,?m(T n S i=1 Si) = n P i=1 m(T Si).?T = n S i=1 Ei,?T Si= ( n S j=1 Ej) Si= Ei,T ( n S i=1 Si) = n S i=1 Ei,?m( n S i=1 Ei) = m(T ( n S i=1 Si) = n P i=1 m(T Si) = n P i=1 mEi. 5.?mE = 0,?E? ?T,T = (E T) (T E),?mT m(E T) + m(T E). ?E T E,?m(E T) mE = 0.T E T,m(T E) mT,? m(E T) + m(T E) mT. 1 ?mT = m(T E) + m(T E),?E? 6.?(Cantor)? ?P?0,1? ? 1 3, ? 2 9, , ?n? 2n1 3n , .?P?0,1? P n=1 2n1 3n = 1(?).? m0,1 = m(P (0,1 P) = mP + m(0,1 P). ?mP = m0,1 m(0,1 P) = 1 1 = 0,?0. 7. A,B Rp?mB +.?A?m(AB) = mA+mBm(AB). ?A? m(A B) = m(A B) A) + m(A B) A) = mA + m(B A). ? mB = m(B A) + m(B A), ?mB +,?m(B A) 0,?G?F,?F E G,? m(G E) ,m(E F) . ?mE 0,?Ii,i = 1,2, ,? S i=1 Ii E,? P i=1 | Ii| mE + .?G = S i=1 Ii,?G?G E,?mE mG P i=1 mIi= P i=1 | Ii| mE + ,?mG mE ,?m(G E) . ?mE = ?E?E = S n=1 En(mEn ), ?En?Gn,?Gn En?m(GnEn) 2n. ?G = S i=1 Gn,G ?G E,?G E = S n=1 Gn S n=1 En S n=1(G n En). ? m(G E) n=1 m(Gn En) 0?G,G E,?m(G E) .? G E = G E = E (G) = E G, ?F = G,?F?m(E F) = m(G E) . 9. E Rq,?An,Bn,?An E Bn?m(BnAn) 0(n ), ?E? 2 ?i, T n=1 Bn Bi,? T n=1 Bn E Bi E.?E Ai,Bi E Bi Ai,? ?i, m n=1 Bn E ! m(Bi E) m(Bi Ai) = m(Bi Ai). ?i ,?m(Bi Ai) 0,?m ? T n=1 Bn E ? = 0.? T n=1 Bn E?Bn? ? T n=1 Bn?E = T n=1 Bn ? T n=1 Bn E ? ? 10. A,B Rp,? m(A B) + m(A B) mA + mB. ?mA = +?mB = +,?mA +?mB 0,?F E,?m(E F) ,?E? ? ?n,?Fn E,?m(E Fn) r?,?rn? ?Ef a = S n=1 Ef rn,?Ef rn?Ef ?f(x)?E? ? ?r,Ef = r?f(x)?E = (,), z?(,) ?x z,f(x) = 3;x 6 z,f(x) =2, ?r,Ef = r = ? ?Ef 2 = z ?f? 2.?f(x),fn(x)(n = 1,2,)?a,b?k? k=1 lim n E ? | fn f | N ? | fn(x) f(x) | 1 k, ? x lim n E ? | fn f | 1 k ? . ?k? x k=1 lim n E ? | fn f | 1 k ? . ?x T k=1 lim n E ?| f n f | 0,?k0,? 1 k0 ,?x lim n E h | fn f | N ?x E h | fn f | 1 k0 i ,?| fn(x) f(x) | 1 k0 limfn. 1 ?E lim n fn= + E lim n fn= E lim n fn lim n fn? ? 4.?E?0,1? f(x) = ( x,x E, x,x 0,1 E. ?f(x)?0,1?| f(x) |? ?f(x)?0 E,?Ef 0 = E?0 6 E,?Ef 0 = E? ?f(x)? ?x 0,1 ?| f(x) |= x?| f(x) |?0,1? 5.?fn(x)(n = 1,2,)?E?a.e.?| fn|a.e.?f. ? 0?c?E0 E,m(EE0) ,?E0?n?| fn(x) | c. ?mE . ?E| fn|= ,Efn6 f?n = 0,1,2, .?E1= Efn6 f( S n=0 E| fn|= ),?mE1= 0.?EE1?fn(x)?f(x).?E2= EE1, ?x E2,sup n | fn(x) | .? E2= k=1 E2sup n | fn| k,E2sup n | fn| k E2sup n | fn| k + 1. ?mE2= lim k mE2sup n | fn| k.?k0?mE2 mE2sup n | fn| k0 .? E0= E2sup n | fn| k0,c = k0.?E0?n,| fn(x) | c,? m(E E0) = m(E E2) + m(E2 E0) c = S n=1 E2n 0,? E E,?m(E E) ?fn?E?f(x).?E0?E?fn? ?, E0 E E(?E?fn?),?mE0 m(E E0) 0,?E E? f(x)?E?m(E E) ,?f(x)?E?a.e.? ?1/n,?En E,?f(x)?En? m(E En) a ( S n=1 Enf a),?f?En?Enf a?m(E0f a) mE0= 0, ?E0f a?Ef a?f?f?En? ? S n=1 En?f(x)a.e.? 9.?fn?E?f,?fn(x) g(x)a.e.?E,n = 1,2, .? f(x) g(x)?E? ?fn(x) f(x),?fni fn,?fni(x)?E?a.e.?f(x).?E0? fni(x)?f(x)?En= Efn g.?mE0= 0,mEn= 0.m( S n=0 En) P n=0 mEn= 0. ?E S n=0 En?fni(x) g(x),fni(x)?f(x),?f(x) = limfni(x) g(x)?E S n=0 En ?f(x) g(x)?E? 10.?E?fn(x) f(x),?fn(x) fn+1(x)?n = 1,2, ,? ?fn(x)?f(x). ?fn(x) f(x),?fni fn,?fni(x)?E?a.e.?f(x).?E0 ?fni(x)?f(x)?En= Efn 0,E| f gn| ( S n=1 En) E| f fn| .? mE| f gn| m( n=1 En) + mE| f fn| = mE| f fn| . 3 ?fn(x) f(x),?0 limmE| f gn| limmE| f fn| = 0?gn(x) f(x). 12.?mE 0,?mE| fn f | 0?0 0,? ?fnk,? mE| fnk f | 0 0 0.(1) ?fnk?f(x)?fnkj ?E?a.e.?f,?mE +,?E?fnkj f(x),?(1)? 13.?mE 0,?k,mE| f | k 5 ?mE| g | k N ?mE| gn g | 0 5,mE| fn f | 0 5 ? ? E| gn| k + 1 E| g | k E| gn g | 1 E| g | k E| gn g | 0. mE| gn| k + 1 mE| g | k + mE| gn g | 0 5 + 5 = 2 5 . ? E h | gnfn gnf | 2 i E| gn| k+1E ? | fn f | 2(k + 1) ? E| gn| k+1E| fnf | 0. ? mE h | gnfn gnf | 2 i mE| gn| k + 1 + mE| fn f | 0 2 5 + 5 = 3 5 . ? E h | fgn fg | 2 i E| f | k+1E ? | gn g | 2(k + 1) ? E| f | kE| gng | 0. 4 ? mE h | fgn fg | 2 i mE| f | k + mE| gn g | 0 0,?N,?n N ?mE| gnfn gf | 0,?e E?me N?men ,? n men Z en | f(x) | dx . ?lim n n men= 0. 4.?mE ,f(x)?E?En= En 1 f n,?f(x)?E? ? P | n | mEn . ?f(x)?E?| f(x) |?E? ?n 1?En?n 1 | f(x) |= f(x) Z E | f(x) | dx = X n=1 Z En | f | dx + X n=0 Z En | f | dx X n=1 (n 1)mEn+ X n=0 | n | mEn = X n=1 | n | mEn+ X n=0 | n | mEn X n=1 mEn= X | n | mEn X n=1 mEn, ?En?E? X n=1 mEn= m( X n=1 En) mE , ? P | n | mEn . ? P | n | mEn,? Z E | f(x) | dx = X n=1 Z En | f | dx + X n=0 Z En | f | dx X n=1 nmEn+ X n=0 | n 1 | mEn = X n=1 | n | mEn+ X n=0 | n | mEn+ X n=0 mEn X | n | mEn+ mE . ?| f(x) |?f+(x)?f(x)?f(x) = f+(x) f(x)? 5.?f(x)?a,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