江泽坚的实变函数答案

? ? Function of Real Variable ? ?410081 mlingxiong mlx0520 2005-11 ? ?1 1.1?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 1.2?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 1.3?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 1.4?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7 ?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8 ?n?12 2.1?Bolzano Weierstrass?. . . . . . . . . . .12 2.2?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13 2.3?. . . . . . . . . . . . . . . . . . . . . . .15 2.4?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16 ?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18 ?19 3.1?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19 3.2?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20 3.3?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22 ?24 4.1?. . . . . . . . . . . . . . . . . . . . . . . .24 2 ?3 4.2Egoroff?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29 4.3?Lusin?. . . . . . . . . . . . . . . . . . . . . . . . .29 4.4?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30 ?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31 ?32 5.1?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33 5.2?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37 5.3Fubini?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42 5.4?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44 ?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45 ?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .47 ?50 ? ? ? ? ? ? ? ? ? ? ?Bernstein?(? ?)? ? 1.1? 1-1-1?(B A) A = B?A B. ?A?B?A B. ?A B?B A B?(B A)A B.?(B A)A B ? 1-1-2?A B = A BC. ?x A B?x A?xB?x A?x BC?x A BC.? x ABC?x A?x BC.?x A?xB?x A B.? 1 ?2 ? 1-1-3?4?(3)?(4)?6?9. ?4(3)?x T B?0 , s.t.xB0?xA0?x T A. ? ?4(4)?x S (AB)?0 ,?x A0B0.?x A0? ?x ( S A) ( S B)?x B0?x ( S A) ( S B).? ?x ( S A) ( S B)?x S A?x S B,?x S A,? ?0 ?x A0?x A0 B0?x S (A B).? ? ?6?x ? T A ?C ?x T A?0 ?xA0?x AC 0 ?x S AC 0 ?x S AC 0 ? ?0 ?x AC 0 ?xA0?x T A?x ? T A ?C .? ? ?9.?An?liminf n An= S n=1 T m=n Am= S n=1 An? limsup n An= liminf n An= S n=1 An?lim n An= S n=1 An. ?An?limsup n An= T n=1 S m=n Am= T n=1 An?liminf n An= limsup n An= T n=1 An?lim n An= T n=1 An. 1-1-4?(A B) S B = (A S B) B?B = . ?B 6= ?x B?x(A S B)B ?x (A B) S B.? 1-1-5?S = 1,2,3,4, A = 1, 2,3,4?F(A)?S = 1 n|n = 1,2, A0 = 1 n|nisodd, A1 = 1,1 3, , 1 2i+1, ?F(A0)?F(A1)? ?3 Solution. F(A) = ,1, 2,3,4,S F(A0) = ,1 n|nisodd, 1 n|niseven,S F(A1) = E|E S. 1-1-6?S?A,?A? A(x) = ( 1,if x A 0,if xA. ?A1,A2, ,An,?S? liminf n An(x) = liminf n An(x), limsup n An(x) = limsup n An(x). Proof. 1).?x liminf n An?1?N,?n N ?x An,?An(x) = 1.?1. ?xliminf n An?0?N,?n N? xAn,?An(x) = 0.?0. 2).? 1-1-7?f(x)?E?a? Ef a = S n=1 Ef a + 1 n, Ef a = T n=1 Ef a 1 n. Proof.?n, Ef a Ef a + 1 n ?Ef a S n=1 Ef a+ 1 n. ?x Ef a,?x E, f(x) a?n? a+ 1 n a+ 1 n ?x S n=1 Ef a+ 1 n ? ?Ef a S n=1 Ef a + 1 n. ? ?Ef a Ef a 1 n, n ?Ef a T n=1 Ef a 1 n. ?x T n=1 Ef a 1 n, ?n, x Ef a 1 n ? ?x E?n,a 1 n a 1 n. ? ?4 1-1-8?fn(x)?E?f(x)? ?a? Ef a= T k=1 liminf n Efn a + 1 k = T k=1 liminf n Efn N ?|fn(x) f(x)| N? x Efn a + 1 k, ?Ef a T k=1 liminf n Efn a + 1 k. ?x T k=1 liminf n Efn a + 1 k, ?k,x liminf n Efn a + 1 k, ? N,?n N?x Efn a + 1 k, ?n N?x E?fn(x) a + 1 k, ?fn(x) f(x)(n ),?f(x) a + 1 k ?k? ?f(x) a.? x Ef a.? T k=1 liminf n Efn a + 1 k Ef a. ? 1.2? 1-2-1?(1, 1)?(,+)?1-1? Solution. y = tan 2x, ?x (1,1). 1-2-2?a b?(a, b) (0,1). Proof.?y = a+(ba)x?x (1,1).?(a, b)?(0,1)?1-1? ?(a, b) (0,1). 1-2-3? ? ? ? ? Proof.?R2?1-2-1? ? ? ?C0?C?(x0,y0)? ?5 ?r? ? : C C0,(x,y) 1 r (x x0,y y0).? ?C?C0? ?C C0. ? ?E?C E?c = C E R2= c?Bernstain?E? 1.3? 1-3-1? ?A. ?Q = r1, ,rn,.? An= (rn,r)|r Q, n = 1,2,?A = S n=1 An.?An?A? 1-3-2? ? ? ? ? ?A?B?B?A? ?A?1-1? 1-3-3? ?n?pn(x) = a0 xn+a1xn1+an1x+ an?n + 1?(a0,a1, ,an)?n ?Pn= pn(x)?n + 1?Rn+1? ?An= (a0,a1, ,an)?An?Pn? An?n = 0?An= a0?n 1?An1 ?An1= b1,b2, ,bm,.?1-3-1? An= S m=1(b m,an)|an Q ? P = S n=1 Pn?Pn? 1-3-4?f(x)?(,+)?f(x)? ?6 ?f(x)?x?f(x0) 6= f(x+ 0),? f(x 0) f(x + 0),?(f(x 0), f(x + 0)?f(x)?x? ?x1 x x2?f(x1) f(x) f(x2),? ? ?f(x)? ?y?1-3-2? ? 1-3-5?A?A A,?A A?A A? Proof. A?1?A?A0= a1,a2,a3, A1= a1,a3,a5,.?A= A A1.?A?7? A A S A1= A,?A A= A1? 1-3-6?A?A? Proof.?A?F? Fn?A?n? ?F = S n=0 Fn?Fn(n 1)? ? ?n = 1?n = k?Fk? Fk= A1,A2,.?Fk+1? Fk+1= m=1 Am a|a A Am, ?Fk+1? ?n?Fn? 1-3-7?A? ?A?. Proof. 1-3-8? I = (x,y,z)|a1 x a2,b1 y b2,c1 z c2 ?a1,a2,b1,b2,c1,c2(a1 a2,b1 b2,c1 c2)?I? ?7 ? Proof. ?E, Q T(c 1,c2) = r1,r2,r3, ? En= (x,y,rn)|a1 x a2,b1 y b2?En An= (x,y)|a1 x a2,b1 0?p,q? q p liminf n An?N?n N ? q p An.?n N,?m,? q p = m n ,?nq = mp,? n?p?n?p = 1,? q p = q Z,?liminf n An= Z. 2.?A,B,C? (1)?A B B A,?A B. (2)?A B?A A S C,?B B S C. Proof. (1)?(A B) T(AT B) = (B A) T(AT B) = ?A = (A B) S(AT B) (B A) S(AT B) = B. (2) C1= C B,?A S C A S C1 A,?A A S C,?Bernstein ?A A S C1.?(B A) T A = (B A) T(AS C1) = ,? B = (B A) A (B A) (A C1) = B C1= B C. ? 1.? 1.1.? 1.2.? 1.3.? 1.4. R1?R1? ? 1.5.? 1.6. R1?(? ?f(x + 0)(?)?) ?10 1.7.? 1.8.?f?(a,b)?x (a,b)|?f (x),f +(x) ? ? 1.9. Rn? 1.10. Rn? 2.? 2.1.? 2.2.? 2.3. Rn 2.4.? 2.5.?R 2.6. a,b? 2.7. a,b? 2.8.? 2.9. R1?Borel 2.10.?f?a,b?a,b?f(x + 0)(?)?f? 2.11. a,b?fn?f? 3.?2c? 3.1. Rn? 3.2. Rn? 3.3. Rn? 3.4. Rn? ?11 3.5. Rn? ?n? ?Rn? ?Rn? ? ?Rn? ? ? ? ?(?)?(?)?(?)?(?)?(?)? ? ? ?Cantor?Cantor? ?0? ? ?R1? ? ? 2.1?Bolzano Weierstrass? 2-1-1?P0 E ? ?P0?N(P,)(?P0? ?)?P0?P 1?E(?P1?).?P0?E? ?P0?N(P,)(?P0?)? N(P,) E. Proof.?P0 E , ? ?P0?N(P,)? ?0, s.t.N(P0,0) N(P,)?N(P0,0)?E? 12 ?n?13 ?N(P,)?E? ?P0?N(P,)?N(P,) E.?0= (P0,P),?N(P0,0) N(P,) E.?P0?E? 2-1-2?Rn= R1?E1?0,1?E1 ?E1. Solution. E1 = E1= 0,1. 2-1-3?Rn= R2?xy?E2= (x,y)|x2+ y2 a? ?x|f(x) a? Proof. x0 x|f(x) a?f(x0) a?f(x)?(,+)? ? = f(x0) a 0? 0?|x x0| a?N(x0,) x|f(x) a?x0?x|f(x) a ?x0? ?x|f(x) a?x|f(x) a ?x|f(x) a = x|f(x) aC? 2-2-3? ?N(P,)?N(P,) = P|(P,P) . Proof. x N(P,)?0= (x,P),?N(x,0) N(P,?x? ? ?N(P,)?N(P,)?(P,P) = ?P? ?N(P,) = P|(P,P) . (?N(P,)?) 2-2-4?Fn(n = 1,2,)? T n=1 Fn= , ?N,? N T n=1 Fn= . Proof(Method 1).?Fn(n = 1,2,)? T n=1 Fn= ,?FC n ? S n=1 FC n = R ,?FC n ?Borel ?FC n1,F C n2, ,F C nk, ? k S i=1 FC ni .? k S i=1 FC ni = ? k T i=1 Fni ?C ,? k T i=1 Fni= .?N = nk,? N T n=1 Fn= . (Method 2).?N, N T n=1 Fn6= ,?xN N T n=1 Fn,?xN. 1). ?xN? T n=1 Fn6= ?2).?xN? ?BolzanoWeierstrass?xN?x0?xN? ?n N?xn FN,?FN?x0 FN,?x0 T n=1 Fn? ? 2-2-5?E Rn, M?E?M? ?N1, ,Nm,?E. Proof.?x E,?Nx M,?x Nx.?Nx?N(x,) Nx,?rx N(x,)?x (x,rx),?N(rx,x) N(x,) Nx ?E S xE N(rx,x),?Rn? ? ?n?15 2-2-10?R1?f(x)? (f,x) = lim 0+ sup |xx| 0, E= x|(f,x) .?x0 E , ? ? 0,?N(x0,) = x|x x0| ?E?x?(f,x) ,? sup |xx0| f(x) inf |xx0| 0 = S n=1 E1 n ,?f(x)?F 2.3? 2-4-1?G? ?(?)?Q = r1, ,rn,.?Q? G?Q = T m=1 Gm?Gm?R1? ?R1= Q S QC= ( S n=1r n) S ? S m=1 GC m ? .?rn? GC m ?Gm= R1?GC m ?R1? ?Baire?(?E Rn?F?E = S n=1 Fn?Fn? ?Fn?E?)?R1? 2-4-3?0,1? ? Proof.?f?0,1? En= x|?x? ?(,),?x1,x2 (,),?|f(x1f(x2)| 1 n ? E = S n=1 En?E?f? ?En?x En , ?x? ?(,),?x ?En? x (,) T En, (,)? x? x En? ?n?16 ?x1,x2 (,),?|f(x1 f(x2)| 1 n ?x En.?En?f ?0,1?E = S n=1 En?2-4-2? ? 2-4-4?R1?c?R1? ?c? Proof. R1?A.A1?R1? ? A1= (,)| (,)| R1 (,)| R1 ,R1. ?A1= c,? : A1 R1. ? ?G A, G 6= ,?G = N S n=1( n,n), ? (n,n)?N?+.?N 0?U = x|(x,E) d?U? ?E U.) Proof. x U?(x,E) 0,?N(x,)?y N(x,), ?n?17 ? (y,E) = inf zE (y,z) inf zE(y,x) + (x,z) = (y,x) + (x,E) 0.?N(P,1), N(Q,2),?1= 1 2(P,F2), 2 = 1 2(Q,F1). ? G1= PF1 N(P,1), G2= QF2 N(Q,2). ?G1,G2?G1 F1, G2 F2.?G1 T G2= . ?P G1 T G2,?P0 F1, Q0 F2?P N(P0,1)? P N(Q0,2),?1 2,? (P0,F2) (P0,Q0) (P0,P) + (Q0,P) 0,?f(P)? ?(P,F1), (P,F2)?P?f(P)?Rn ?0 f(P) 1?F1?f(P) 0?F2?f(P) 1. ? ? ?Lebesgue?Lebesgue? ?Caratheodory? ? ?Lebesgue? ? ? ?Lebesgue? ? 3.1? 3-1-1?E?mE . ?E I Rn,?I?mE mI 0?ei?ei? ?Ii?|Ii| 0?x E? ? 0?m(E T N(x,) 0. Proof.?x E, x 0,?m(E T N(x,x) = 0.? S xE N(x,x) E,?N(x,x)xE?E?2-2-5? ?E,?N(x1,1), ,N(xn,n),? S n=1 N(xn,n) E. ? mE = m( n=1 (E N(xn,n) X n=1 m(E N(xn,n) = 0. ? 3.2? 3-2-3?6?mT = +?T? Solution.?En= (n,+),n = 1,2, , T = R,?E = T n=1 En= ,?m(T T E) = 0.?m(T T En) = mEn= + +. 3-2-4? ?A?B?m(A B) + m(A B) = mA + mB. ?21 ?A B = A (B A),?A (B a) = , (B A) (A B) = ? m(A B) + m(A B)= m(A (B A) = mA + m(B A) + m(A B) = mA + mB. 3-2-5? ?E Rn? ? 0?Rn?G?G E? mG mE + . Proof.? 0,?In? S n=1 In E? X n=1 |In| mE+.?G = S n=1 In,?G E?mG = m ? S n=1 In ? X n=1 |In| mE+. 3-2-6?Rn? ?Ek? m(liminf k Ek) liminf k m(Ek). ?k0?m( S k=k0 Ek) +? m(limsup k Ek) limsup k m(Ek). ? T m=k Em?5? m(liminf k Ek)= m( S k=1 T m=k Em) = m(lim k T m=k Em) = lim k m( T m=k Em) liminf k m(Ek). ? S m=k Em?6? m(limsup k Ek)= m( T k=1 S m=k Em) = m(lim k S m=k Em) = lim k m( S m=k Em) limsup k m(Ek). 3-2-7?E?Rn?mE +.?E?Ek ? X k=1 mEk 0?G E?F E?m(GE) , m(E F) 0? ?In?E?E S n=1 In? X n=1 |In| mE +.?G = S n=1 In. ?23 ?mG X n=1 |In| mE + .?G?mE ?(3-1-1?)? m(E G) = mE mG . (2)?E?E?E = S n=1 En?(1)?En?Gn?En Gn?m(GnEn) 2n . ?G = S n=1 Gn?E = S n=1 En S n=1 Gn= G?m(GE) = m ? S n=1 Gn S n=1 En ? = m ? S n=1(G n En) ? X n=1 m(Gn En) 0? ?G?G EC?m(G EC) ?F = GC?F? F = GC (EC)C= E?m(EF) = m(E T(GC )C) = m(E T G) = m(GEC) . ? 3-3-5?Rn? ?En, n = 1,2,?E1 E2 En , ? m n=1 En ! = lim n mEn. Proof.?En?mEn= +,? ?mEn 0,?Gn En? mGn a?E1?E2? ?Ef a = E1f aE2f a?f? ?E1 E2? 4-1-3?mE 0? ?F E?m(E F) n.? 0, n,m(An) 2. ? m(An) mE 0, n0,m(An0) 2. ?26 ?E1= E0 f(x) n0 = E An0?3-3-2? F E1?m(E1 F) 0,? X n=1 mEfn 0(n ),?lim n mEfn = mE,?x lim n Efn ,?Nx? n Nx?x Efn ,?fn(x) ,?fn(x) 0(n ).? ?4-2-1?m