周民强《实变函数论》课后答案

����� ������������� ��� �� Ex 1. � {fj(x)} ������� Rn ����������ffflfi�ffi��� {x : fj(x) ≥ 1 k }, (j, k = 1, 2, · · ·) !�" �� { x : lim j→∞ fj(x) > 0 } . #�$ { x : lim j→∞ fj(x) > 0 } = ∞⋃ k=1 { x : lim j→∞ fj(x) ≥ 1 k } = ∞⋃ k=1 { x : lim n→∞ sup j≥n fj(x) ≥ 1 k } = ∞⋃ k=1 ∞⋂ n=1 ∞⋃ j=n { x : fj(x) ≥ 1 k } Ex 2: � {fn(x)} ������� [a, b] ����������ff E ⊂ [a, b] %�& lim n→∞ fn(x) = χ[a,b]\E(x), x ∈ [a, b]. '�( En = {x ∈ [a, b] : fn(x) ≥ 1 2}, fi�)� �* lim n→∞ En . #�$,+ lim n→∞ fn(x) = χ[a,b]\E(x) = { 1, x ∈ [a, b] \E 0, x ∈ E -/. x ∈ limn→∞En, 0/1 ∃N , 2 n ≥ N 3 ff x ∈ En, 1 fn(x) ≥ 1/2, 4/5 [a, b] \ E ⊂ limn→∞En; . x ∈ lim n→∞ En, 6 ∀n, ∃k ≥ n, 7�8 fk(x) ≥ 1/2, 9�: E ∩ lim n→∞ En = ∅. ;�<�=�> ? lim n→∞ En = [a, b] \E. Ex 4: � f : X → Y, A ⊂ X, B ⊂ Y , fi�@�A���B�C�D�E�FHG (i)f−1(Y \B) = f−1(Y ) \ f−1(B); (ii)f(X \A) = f(X) \ f(A). #�$ (i) :�I�J�K�L (ii) M�N�O�M�P�J�K ffflQ�R�S�T f(x) = x2, X = [−1, 1], Y = [0, 1], A = [0, 1]. Ex 5: fi�U�V�W {(x, y) : x2 + y2 < 1} X�Y W {(x, y) : x2 + y2 ≤ 1} Z�[ ��\�\�]�^ L #�$fl_�`ba {(x, y) : x2 + y2 < 1} c #�d ⋃ r∈[0,1) {(x, y) : x2 + y2 = r}, _�eba {(x, y) : x2 + y2 ≤ 1} c #�d ⋃ r∈[0,1] {(x, y) : x2 + y2 = r}, f S�T [0, 1) g [0, 1] h�i�O�j�k�K�M�M�l�m�L 1 ????? www.ucourse.net（??：??????：ucourse） ? 1 ?，? 19 ? 6/10/2009 Ex 7: � f(x) ������� [0, 1] ����n�o�����ff %�p���q � M , r�s ]�t [0, 1] u�v�w�& x�y � $ x1, x2, · · · , xn, z�& |f(x1) + f(x2) + · · ·+ f(xn)| ≤ M, fi�{�|� �* E = {x ∈ [0, 1] : f(x) 6= 0} ��} �� L ~b $ S�T E = {x ∈ [0, 1] : f(x) 6= 0} = ∞⋃ k=1 {x : f(x) > 1 k } ⋃ ∞⋃ k=1 {x : f(x) < − 1 k } +�€ -��‚�ƒ�- l�„�… k, †�‡ {x : f(x) > 1 k } g {x : f(x) < − 1 k } ˆ�‰ ?�Ł ‡�L Ex 9: � E � R3 u ���� �ff % E u � v�w�‹ ����Œ��Ž ��&� ��fffi�{�| E ��} � L ~‘ $ „/’ x0 ∈ E, 6 E ⊂ ⋃ r∈Q+ S(x0, r), “‘” Q + •/–/—/˜ ? c/™/‡ ff S(x0, r) •/– 5 x0 d ”›š r d�œ��ž�Ÿ�  ff¢¡ …�£�:�¤ d�¥�¦�§ ff 9�:�¨�© ~ lbª›P ž r,S(x0, r)∩E ‰ ¥ ™�‡ fffl« S(x0, r) ∩ E ⊂ ⋃ r′∈Q+ C(x1, r ′), “b” C(x1, r ′) •�– 5 x1 d ”›š r′ d�œ��žba ff x1 ∈ S(x0, r) ∩ E, ¬/M/­ C(x1, r′) ∩ E ⊂ ⋃ r′′∈Q+ C(x1, r ′) ∩ C(x2, r′′), x2 ∈ C(x1, r′) ∩ E, « C(x1, r ′) ∩ C(x2, r′′) ®�¯ ? E ž�°�± †�L Ex 12: � E = ∞⋃ n=1 An, ' E = c, fi�{�| p�� n0, r�s An0 = c. ~b $ S�T�² ?�³�´�µ ™ ž †�‡ [0, 1]∞ = {(x1, x2, · · · , xn, · · ·) : xn ∈ [0, 1]}, 6 ∞⋃ n=1 An ∼ [0, 1] ∞. O�¶�· An ¸ O�¹�º fffl» 6�¼ _ E •�– d�½�¾�ž O�º § Lfl· f : ∞⋃ n=1 An → [0, 1] ∞ ‰�M�M�¿�À ff · f(An) = Dn ⊂ [0, 1]∞, 6 An ∼ Dn. S�T [0, 1]∞ ” ž †�‡ Xn = {(0, 0, · · · , 0, xn, 0, · · ·) : xn ∈ [0, 1]} 5�Á�¿�À Pn : Dn → Xn, Pn((x1, x2, · · · , xn, · · ·)) = (0, 0, · · · , 0, xn, 0, · · ·) +Â Xn žÃµ ™ d c, 9Ã: RÃÄ Dn < c, 6 Pn(Dn) ≤ Dn < c, 9Ã: ∀n, ∃x∗n, 7Ã8 (0, 0, · · · , 0, x∗n, 0, · · ·) /∈ Pn(Dn), Å « (x1, x2, · · · , x∗n, · · ·) /∈ Dn, Æ (x ∗ 1, x ∗ 2, · · · , x ∗ n, · · ·) /∈ ∞⋃ n=1 Dn = [0, 1] ∞, Ç�È ff 9�:�É�Ê n0, 7�8 Dn0 = c, Å « An0 = c. Ex 13: � f(x) ������� R1 ����Ë�Ì���Í�����ffflfi�{�|��� E = {x : ]�t v�w � ε > 0 & f(x + ε)− f(x− ε) > 0} � R1 u � Y L 2 ????? www.ucourse.net（??：??????：ucourse） ? 2 ?，? 19 ? 6/10/2009 ~b $ S�T E ž�Î ‡ Ec, ƒ�- Ec = {x : É�Ê ε > 0 ? f(x + ε)− f(x− ε) ≤ 0} ;�<�Ï�™ ž�Ð�Ñ�Ò�Ó�Ô f(x + ε)− f(x− ε) ≥ 0, 9�: Ec = {x : É�Ê ε > 0 ? f(x + ε)− f(x− ε) = 0} l�„�… ž x0 ∈ E c, 6�É�Ê ε > 0 7�8 f(x0 + ε)− f(x0− ε) = 0, ( Õ�Ö Ò�×�Ø  Ï�™ f Ê�Ù�Ú (x0−ε, x0+ε) ”Û‰ÝÜÝÞÝÏݙ ) 9Ý: ∀y ∈ (x0−ε, x0+ε), ¥ 5ݒ δ = min{y−(x0−ε), (x0+ε)−y)}, ? f(y + δ)− f(y− δ) = 0, 9�: (x0 − ε, x0 + ε) ⊂ Ec, Å « x0 ‰ Ec žbß † ff Æ Ec d�` ‡ ff 0�1 E ‰ e ‡�à Ex 14: � F ⊂ Rn ��&�á�Y �ff E � F ��\ y�â�x�ã �fffi�{�| E′ ∩ F 6= ∅. ä�Z ff ' F ⊂ Rn % ]�t F � v \ â�x�ã E, & E′ ∩ F 6= ∅, fi�{�| F ��&�á�Y L ~‘ $ · F ⊂ Rn ‰ ?/å/e ‡ ff E ‰ F ž M ±/æ/Ł/ç ‡ ff 9/: E ‰ ?/å/æ/Ł ‡ ff + Bolzano-Weierstrass P�c - E ′ 6= ∅, 9�: E′ ⊂ F ′ ⊂ F , Å « E′ ∩ F = E′ 6= ∅. è h/L . F ⊂ Rn é l  F ž „/M æ/Ł/ç ‡ E, ? E′ ∩ F 6= ∅, . F æ/å ff 6/ê/ë ¥ 5ÃìÒÃM ±ÃæÃí † ¦ 7ÃîÃï ?ÃðÃŁ † ffflñ ¼Ã„Ã’ x1 ∈ F , 6 O(x1, 1) òÃó ? F ž † ff ’ x2 ∈ O(x1, 1) c ∩ F , f S�T O(x1, max{2, d(x1, x2)}), ×�± Ù�Ú�ò�ô ? F ž † fföõ :�ì�’�M ± O/÷�ø ž�æ�í † ¦ x1, x2, · · ·, ’ E = {x1, x2, · · ·}, 6 ? E′ = ∅, Å « E′ ∩ F = ∅, g �‚ Ç È ff 9�: F ‰ ?�å ‡�L ù M/ú   l/„/… x0 ∈ F ′, 6/É/Ê F ” ž/æ/í † ¦ xn 7/8 xn → x0, ’ E = {xn : n = 1, 2, · · ·}, + �‚ E′ ∩ F 6= ∅ 1�8 x0 ∈ F , Å « F 0�‰ e ‡�L Ex 15: � F ⊂ Rn ��Y �ff r > 0, fi�{�|��� E = {t ∈ Rn : p�� x ∈ F, d(t, x) = r} ��Y L ~b $ · t0 ∈ E′, 6�É�Ê E ” ž ¸�û † ¦ tn, 7�8 tn → t0, + tn ∈ E - É�Ê xn ∈ F 7 8 d(tn, xn) = r. . xn ¨�ü ?�Ł�± O�ý ž † ff 6�Å�þ ± N `�ß ff xn ‰�Ü���† ¦ ff O�¶�� d xk0 , 6 + d(tn, xk0) = r, (n ≥ N) K�1 - d(t0, xk0) = r, 4�5 t0 ∈ E. » 6 xn ü æ Ł ± O ý ž † ff + d(xn, t0) ≤ d(xn, tn)+d(tn, t0) = r+d(tn, t0), ; < tn → t0, - xn ‰ ?�å�æ�í���¦ ff + Bolzano-Weierstrass P�c ff É�Ê�÷�ø ž�ç�¦ ff O/¶���� xn ��� ÷�ø ff · xn → x0, 6 + F ‰ e ‡ - x0 ∈ F , é d(t0, x0) = r. ( S�T�� Â� � �ž���� O ½ I ? ) d(x0, t0) ≤ d(x0, xn) + d(xn, tn) + d(tn, t0) 5�Á d(x0, t0) ≥ d(xn, tn)− d(x0, xn)− d(tn, t0). ¡ $ �� ½�¾� . F ⊂ Rn ‰ e ‡ ff r > 0, 6 ⋃ x∈F S(x, r) ‰ e ‡ ff “b” S(x, r) ‰�5 x d ”›š r d�œ��ž�Ÿ�  L Ex 18: � f ∈ C(R1),{Fk} � R1 u �������� ���ffflfi�{�| f ( ∞⋂ k=1 Fk ) = ∞⋂ k=1 f(Fk). ~b $,+ ¿���‡ ž M�N Ô�� f ( ∞⋂ k=1 Fk ) ⊂ ∞⋂ k=1 f(Fk) ����J�K�L ù M�ú   ff l y0 ∈ ∞⋂ k=1 f(Fk), 6�l�„�… k, É�Ê xk ∈ Fk 7�8 f(xk) = y0. . xk � ü ?�? Ł�± ¸ O�¹�ý ž † ff 6�É�Ê k0, 2 k ≥ k0 3 xk = xk0 , Å « y0 = f(xk0) ∈ f ( ∞⋂ k=1 Fk ) . . xk 3 ????? www.ucourse.net（??：??????：ucourse） ? 3 ?，? 19 ? 6/10/2009 ü ? æ Ł ± ¸ O ¹ ý ž † ff +�� Ô ( ? å e ), † ‡ {xk} ®�� ? M ±�� † x0, ��� x0 ∈ ∞⋂ k=1 Fk, f + f ž�³�´�Ô - f(x0) = y0, Å « y0 ∈ f ( ∞⋂ k=1 Fk ) . 4�5 f ( ∞⋂ k=1 Fk ) ⊃ ∞⋂ k=1 f(Fk). ff�<�= >�1�8 f ( ∞⋂ k=1 Fk ) = ∞⋂ k=1 f(Fk). Ex 21: � E ⊂ Rn. ' E 6= ∅, % E 6= Rn, fi�{�| ∂E 6= ∅. ~‘ $ ( �fi ½/¾/ ~‘ Rn ”ffifl `fi�/e/ž ‡/< �fi ‰ ∅ ! Rn) . ∂E = ∅, 6 Rn = E◦ ∪ (Ec)◦ Å « E fl `���e L · E ⊂ Rn, E 6= ∅, é E fl `���e ff#" ~ E = Rn. . Ec 6= ∅, 6�É�Ê x0 ∈ E c, 9 d E fl `���e ff 4�5 Ec 0�fl `���e ff Å « x0 ‰ E c žbß † ff 6�É�Ê δ0 > 0 7�8 O(x0, δ0) ⊂ Ec, � δ∗ = sup O(x0,δ)⊂Ec δ . δ∗ = +∞, 6 Ec = Rn, E = ∅ g E 6= ∅ Ç/È ff 9/: δ∗ < +∞, �‘+  Ec ‰ e ‡ ff 9/: e/Ÿ O(x0, δ ∗) ⊂ Ec S/T Ÿ/  S(x0, δ ∗) Ò/ž † x ∈ Ec, + Ec ‰ ` ‡ ff x ‰ ß † ff Æ/É/Ê δx > 0, 7�8 O(x, δx) ⊂ Ec, « S(x0, δ ∗) ⊂ ⋃ x∈S(x0,δ∗) O(x, δx) 5�Á S(x0, δ ∗) ‰ ?�å�e ‡ ff + Heine-Borel ?�Ł�ç�$�% P�c ff É�Ê ?�Ł�± xi 7�8 S(x0, δ ∗) ⊂ m⋃ i=1 O(xi, δxi) ⊂ E c � δ′ = min{δxi , i = 1, 2, · · · , m} 6 O(x0, δ ∗ + δ′) ⊂ Ec ff × g δ∗ ž P�&�Ç�È ff Ø  Ec = ∅, 1 E = Rn. Ex 22: � G1, G2 � R n u�'�(�)�* ��V� �ffflfi�{�| $ G1 ∩G2 = ∅. ~b $ ( ú�+�M�,�-�+ ) l „ … x0 ∈ G1, +ö G1 ‰ ` ‡ ff 9 : É Ê δ0 > 0, 7 8 O(x0, δ0) ⊂ G1, � +ö G1∩G2 = ∅, 9�: O(x0, δ0) ∩G2 = ∅, 0�1 x /∈ G2, 4�5 G1 ∩G2 = ∅. ( ú�+�. è ~ + ) . G1 ∩G2 6= ∅, ’ x0 ∈ G1 ∩G2, +›Â G1 ‰ ` ‡ ff 9�:�É�Ê δ1 > 0, 7�8 O(x0, δ1) ⊂ G1, �b+ x0 ∈ G2, 9�:�l�„�… δ > 0, ? O(x0, δ) ∩G2 6= ∅, /�0�1 S�T δ = δ1, ? O(x0, δ) ⊂ G1 ∩G2 6= ∅. × g € -��‚ ¹�Ç�È ff 9�: G1 ∩G2 = ∅. �� �2 †�3 – $ x ∈ E ⇔ ∀δ > 0, O(x, δ) ∩ E 6= ∅. Ex 23: � G ⊂ Rn. ' ] v�w � E ⊂ Rn, & G ∩ E ⊂ G ∩ E, fi�{�| G � V� L ~b $ ( ú�+�M $ ,�-�+ ) 4 ????? www.ucourse.net（??：??????：ucourse） ? 4 ?，? 19 ? 6/10/2009 ’ E = Gc, + �‚ G ∩ E ⊂ G ∩ E - G ∩Gc ⊂ G ∩Gc = ∅ = ∅ 9�: Gc ⊂ Gc Å « Gc ‰ e ‡ ff 0�1 G ‰ ` ‡�L ( ú�+�. $ è ~ + ) . G O�‰ ` ‡ ff 6�É�Ê x0 ∈ G,x0 O�‰ G žbß † ff 1�l�„�… δ > 0, O(x0, δ) ∩Gc 6= ∅. S T δ1 = 1, ’ x1 ∈ O(x0, δ1) ∩Gc, S�T δ2 = min{1/2, d(x1, x0)}, ’ x2 ∈ O(x0, δ2) ∩Gc, õ : 4�5 ff ¥ 5�8�£�† ¦ xn, ? xn → x0, xn ∈ Gc, ’ E = {x1, x2, · · · , }, 6 G ∩ E = ∅, 6 G ∩ E ®���ü ? † x0, × g �‚ Ç�È ff Ø  G ‰ ` ‡�L ¡ $87 Õ Ò�?�$ · G ⊂ Rn, 6 G ‰ ` ‡ ž�9�: ó 2 �‚ ‰�l�„�… ž E ⊂ Rn, ? G∩E ⊂ G ∩ E. ~ $ · G ⊂ Rn ‰ ` ‡ ff l�„�… ž E ⊂ Rn, · x0 ∈ G ∩E, 6 x0 ∈ G, x ∈ E, + x ∈ E - É�Ê E ž † ¦ xn 7�8 xn → x0, « x0 ∈ G, 9�:�É�Ê δ > 0, 7�8 O(x0, δ) ⊂ G, 9�: n 9�: ;�< xn ∈ E ∩O(x0, δ), 4�5 x0 ∈ G ∩ E, Æ G ∩ E ⊂ G ∩ E. Ex 24: � a, b, c, d � n���ff % P (x, y) = ax2y2 + bxy2 + cxy + dy. fi�@��� {(x, y) : P (x, y) = 0} &�= ��FHG # $ . {(x, y) : P (x, y) = 0} É Ê ß † ff · (x0, y0) ‰ “ ß † ff 6 l ªöP ž y Þ y0,P (x, y0) = ax2y20+bxy 2 0+cxy0+dy0 > d �  x ž .�? Ï ™ ff ó�@ Ê (x0, y0) ž þ ± δ > 0 Ù Ú ”�A d�B ff 9 : ? ay20 = 0, by 2 0 +cy0 = 0, dy0 = 0, Ê (x0, y0) ž δ > 0 Ù Ú ” f „ ’ M † (x0, y1), y1 6= y0, ýfiC ? P (x, y1) = ax 2y21 + bxy 2 1 + cxy1 + dy1 > d �  x ž .fi?/Ï/™/Ê/M ± x ž Ù/ÚfiA d B ff 9�: ? ay21 = 0, by 2 1 + cy1 = 0, dy1 = 0, ff�<�=�>�ó�@ a = b = c = d = 0, ��� × 3�4�D E ž †�‡�‰ R2, F�G ×�H /�I�J�K�4�D ž †�‡�O ¥ ?bß †�L Ex 25: � f : R1 → R1, ( G1 = {(x, y) : y < f(x)}, G2 = {(x, y) : y > f(x)}. fi�{�| f ∈ C(R1) L�%�M�L G1 X G2 � V� L ~b $ · f ∈ C(R1), ê�ë�¨ ~b G1 ` ff G2 ¥�4�N ~b Lfl„�’ (x0, y0) ∈ G1, 6 + G1 ž P�& - y0 < f(x0), · ε = f(x0)−y0 2 , + f Ê x0 ž�³�´�Ô ff É�Ê δ > 0, 7�8�2 x ∈ O(x0, δ) 3 ff |f(x) − f(x0)| < ε, Å « Ï/™ f(x) Ê/Ù/Ú O(x0, δ) ž Ï/™fiOfiPfiQ/Ê/5 (x0, f(x0)) d ” š ff x úSR�T�U d 2δ,y úSR�T�U d 2ε ž�V K h ”öLW� δ0 = min{ε, δ}, 6 O((x0, y0), δ0) ⊂ G1, 9�: (x0, y0) ‰ G1 žbß † ff Æ G1 ‰ ` ‡�à ù MÝú   . G1 , G2 X ‰ ` ‡ ff l݄݅ x0 ∈ R1, „Ý… ε > 0, SÝT (x0, f(x0)+ε) ∈ G2, 6ÝÉÝÊ δ1, 7 8 5 (x0, f(x0)+ε) d ”öš 2δ1 d T�U ž�Y�Z ú�K�[ ± ü Ê G2 ”�\ S T (x0, f(x0)−ε) ∈ G1, 6ÃÉÃÊ δ2, 7Ã8Ã5 (x0, f(x0) − ε) d ”š 2δ2 d T]U ž]Y]Z ú]K][ ± üÃÊ G1 ” ff ’ δ = min{δ1, δ2}, 6 2 x ∈ O(x0, δ) 3 ff |f(x)−f(x0)| < ε, 9 : f(x) Ê x0 † ³ ´ ff + x0 ∈ R 1 ž „�… Ô - f(x) ∈ C(R1). ¡ � $#^ ‡�< ž�Ô�� è ¿�Ï�™ ž�³�´�Ô�ž�4�N�_ �` ? 1: · f(x) P�&�Ê Rn Ò ff 6 f ∈ C(Rn) ž�9�: ó 2 �‚ ‰ $ l�„�… ž t ∈ R1, †�‡ E1 = {x ∈ R n : f(x) ≥ t}, E2 = {x ∈ R n : f(x) ≤ t} X ‰ e ‡�L ( a�b P36 Q 2) 5 ????? www.ucourse.net（??：??????：ucourse） ? 5 ?，? 19 ? 6/10/2009 2: · f(x) P�&�Ê Rn Ò ff 6 f ∈ C(Rn) ž�9�: ó 2 �‚ ‰ $ l�„�… ž�` ‡ G ⊂ R1, c�� ‡ f−1(G) ‰ Rn ” ž�` ‡�L 3. · f(x) P�&�Ê e�d i [a, b] Ò ff . †�‡ Gf = {(x, f(x)) : x ∈ [a, b]} ‰ R2 ” ž�� ‡ ff 6 f ∈ C[a,b]. Ex 26: fi�@�e R1 u ��\�f�V� �g�D��� �h���i�� ��j�k G #�$ · En = {(a1, a2, · · · , an) : a1, a2, · · · , an ∈ R1, a1 ≤ a2 ≤ · · · ≤ an}, ƒ ~ En ž�µ ™ d c, � E = ∞⋃ n=1 En, Æ E ž�µ ™�0 d c. · F = {(a1, a2, · · · , an, · · ·) : a1, a2, · · · ∈ R1, a1 ≤ a2 ≤ · · · ≤ an ≤ · · ·}, F ž�µ ™�0 d c. l R1 ž „�… ` ‡ G, +›` ‡ ž ;�l�P�c ff G ‰ ¥ ™ ± ¸ O�¹�º ž�`�d i ž�§ ffnm�o ±�`�d i�p�†�l�J ž M ± Å Y £ ;�ž�q�¦�_ ‰ E ! F ” ž M ± † ff 9�: R1 ” ž M�r ` ‡�l�J ž ‡�s ž�µ ™�O�t�u c. ù M�ú   „�…�Õ�™ x ∈ R1,(x, x + 1) ‰�M ±�` ‡ ff × C���K�1�8�£/‡�s ž/µ ™/O Y� c. ff�<�=�>�� ? R1 ” ž M�r ` ‡�l�J ž ‡�s ž�µ ™ d c. ¡ $#7 Õ Òb+›` ‡ 4�v J ž σ− w�™ Borel ‡ 4�ž�µ ™ d c, 9�: R1 ž M�r ` ‡�s ��� µ ™�O�t�u c. Ex 27: � Fα � R n u ��\�h &�á�Y �ff ' v�x�y�u�& x�y $ Fα1 , Fα2 , · · · , Fαm ff¢Ž & m⋂ i=1 Fαi 6= ∅, fi�{�| ⋂ α Fα 6= ∅ . ~b $ . ⋂ α Fα = ∅, 6 ⋃ α F cα = R n, „�’�M ± Fα0 , ��� Fα0 ⊂ ⋃ α F cα, 0�1 F c α ‰�l ?�å�e ‡ Fα0 ž M ±�`�$�% ff + Heine-Borel ?�Ł�ç�$�% P�c ff É�Ê ?�Ł�± F cαi , i = 1, 2, · · · , l, ? Fα0 ⊂ l⋃ i=1 F cαi 9�: Fα0 ⋂( l⋂ i=1 Fαi ) = ∅, × g € -��‚ ¹�Ç�È ff Æ ⋂ α Fα 6= ∅. Ex 28: � {Fα} � Rn u � &�á�Y �h�ff G � V� %�&⋂ α Fα ⊂ G, fi�{�| {Fα} u�p���& x�y Fα1 , Fα2 , · · · , Fαm ff r�s m⋂ i=1 Fαi ⊂ G. ~b $,+ ⋂ α Fα ⊂ G, - Gc ⊂ ⋃ α F cα, „�’�M ± Fα0 , ��� Fα0 ⋂ Gc ⊂ ⋃ α F cα, × 3 Fα0 ⋂ Gc d ? å e ‡ ff F cα ‰ l î ž M ± `�$�% ff + Heine-Borel ? Ł ç�$�% P c ff É Ê ? Ł ± F cαi , i = 1, 2, · · · , l, ? Fα0 ⋂ Gc ⊂ l⋃ i=1 F cαi = ( l⋂ i=1 Fαi )c 6 ????? www.ucourse.net（??：??????：ucourse） ? 6 ?，? 19 ? 6/10/2009 9�: l⋂ i=1 Fαi ⊂ G ∪ F c α0 Å « l⋂ i=0 Fαi ⊂ G ∩ Fα0 ⊂ G. Ex 29: � K ⊂ Rn ��&�á�Y �ff {Bk} � K ��V�z�{�|�ffflfi�{�| p�� ε > 0, r K � v \���} u�~ ff ε }��€���z��‚�t {Bk} u ��\ y L ~  $ K ⊂ Rn ‰ ? å e ‡ ff {Bk} ‰ K ž ` Ÿ�$�% ff + Heine-Borel ? Ł�$�% P c - ff É�Ê ?�Ł�±�`�Ÿ {Bk}, (k = 1, 2, · · · , m), 7�8 K ⊂ m⋃ k=1 Bk. 9/: ∀x ∈ K, É/Ê/þ ±/`/Ÿ Bk0 , 7/8 x ∈ Bk0 , +  Bk0 ‰ `/Ÿ ff 9/:/É/Ê δx > 0, 7/8 O(x, δx) ⊂ Bk0 . S�T £ K ⊂ ⋃ x∈K O(x, δx/2) f ?�ƒ ^ Heine-Borel ?�Ł�$�% P�c ff É�Ê ?�Ł�± xi(i = 1, 2, · · · , l) 7�8 K ⊂ l⋃ i=1 O(xi, δxi/2) (1) ’ ε = min 1≤i≤l δxi 2 , 6�: ε > 0 „�… ” ž 2�† L l�„�… x ∈ K, S�T 5 x d ”›š ff ε d�œ� ž�Ÿ B(x, ε), ñ ¼ + (1) - É�Ê�þ ± i0, 7�8 x ∈ O(xi0 , δxi0 /2), “�? × 3�l�„�… y ∈ B(x, ε), ? d(y, xi0) ≤ d(y, x) + d(x, xi0 ) ≤ ε + δxi0 /2 < δxi0 , 0�1 B(x, ε) ⊂ O(xi0 , δxi0 ). « O(xi0 , δxi0 ) ‡�ˆ�ì�’�‰�[ ± ü�Ê�þ ±�`�Ÿ Bk ”›L Ex 32. fi�{�| R1 u � } ��‰�Ł� (�� Gδ L ~b $ ( �� ž ~b 4�N� R1 ” ž�? c�†�‡�O�‰ Gδ ‡ ) · E = {a1, a2, · · · , an, · · ·} ‰ R1 ” ž/¥ ™fi‹fiŒ/‡ ff . E ‰ Gδ ‡ ff 6/É/Ê ¥/¦/±/` ‡ Gi, i = 1, 2, · · ·, 7�8 E = ∞⋂ i=1 Gi, 9�: ? R1 = Ec ∪E = ( ∞⋃ i=1 Gci )⋃( ∞⋃ i=1 {ai} ) +›Â E ⊂ Gi 9�: Gi ‹�Œ ff Æ Gci ‰ æbß † ž�e ‡ ffö« Ð †�‡�‰ æbß † ž�e ‡ ff 9�: R1  • – d ¥ ¦ ± æ,ß † ž e ‡ ž § ff + Baire P c - R1 æ ß † ff × M Ç È Ø  E O�‰ Gδ ‡ L 7 ????? www.ucourse.net（??：??????：ucourse） ? 7 ?，? 19 ? 6/10/2009 Ex 34. fi�{�| � [0, 1] � (�Ž����� A���� f(x): ��&� �����‘�ff � ⠏ � ( �‘ L � © ¡ …�£ [0, 1] Ò P�& ž Ï�™ f(x) ž�³�´ †�‡�l�J Gδ ‡ ffö« ý�C ¥ 5 ~b [0, 1] ” ž ? c�™�‡�0�O�‰ Gδ ‡�L Ex 36. � E ⊂ R1 ��’�“�} �� L ' E â�” E���ffflfi�{�| E \E � E u ‰�Ł L ~  $ E ⊂ R1 ‰ —�• ¥ ™ ‡ ff +ö E æ�– K † ff 9 : E ó ‰ ¥ ¦ ‡ ff « é E = E′∪E = E′,( ï ?�– K�† Ø  4 ? † X ‰ � † ), «�é O ¥ E′ = E, » 6 E ‰�—�˜�‡ ( —�™�‡ ), «�— • —�™�‡ ž�µ ™ d c, 9�: E \E = E′ \E 6= ∅. © ~ E \E ⊃ E. ∀x0 /∈ E \E, ∃δ0 > 0, 7�8 O(x0, δ0)∩ (E \E) = ∅, +›Â E ⊂ E, 9�: ff O(x0, δ0)∩E = ∅, ! O(x0, δ0) ⊂ E, « <�š ‰�O ¥ ž ( 9 E ¥ ™ ), Æ O(x0, δ0) ∩E = ∅, 9�: x0 ∈ E c . Ex 37. fi�{�| � Rn u � v \ Y �›�} Gδ �ff v \�V� �›�} Fσ L ~b $Ý+›Â�` ‡ ž�œ ‡�‰ e ‡ ff Gδ ‡ ž�œ ‡�‰ Fσ ‡ ff 9�:�ê�ë�¨�© ~bž œ�Ÿ�: LÛ· F ‰ Rn ” ž�e ‡ ffflS�T � Ï�™ f(x) = d(x, F ), 5�Á F = {x : d(x, F ) = 0} = ∞⋂ k=1 {x : d(x, F ) < 1 k } +ž � Ï�™ ž�³�´�Ô - {x : d(x, F ) < 1 k } ˆ d�` ‡ ff 9�: F ‰ Gδ ‡�L Ex 38. � f(x) : [0, 1] 7→ [0, 1], ' �� Gf = {(x, f(x)) : x ∈ [0, 1]} � [0, 1]× [0, 1] u � Y �ff#  f ∈ C([0, 1]). ~‘ $ Å /‚/- Gf = {(x, f(x)) : x ∈ [0, 1]} ‰ ?/å/e ‡ ( � ‡ ), l/„/… x0 ∈ [0, 1] 5 Á/™ ¦ xn → x0,(xn, f(xn)) ∈ Gf , 9/: f(xn) ‰ ?/å ™ ¦ ff Å « É/Ê/÷/ø ç/¦ f(xnj ), · f(xnj ) → y0, 6 (xnj , f(xnj )) → (x0, y0), +ö Gf ‰ e ž ff 9 : (x0, y0) ∈ Gf , 9 : y0 = f(x0), 4�5�l�„�…�™ ¦ xn → x0 É�Ê ç�¦ f(xnj ) → f(x0), Å « f(xn) → f(x0)( ¡�;�c�6 ). Æ f Ê x0 ³�´ ff Å « f ∈ C([0, 1]). Ex 41. � F1, F2, F3 � R n u�¢ y '�(�)�* � Y �ffflfi�U f ∈ C(Rn), r�s (i) 0 ≤ f(x) ≤ 1; (ii) f(x) = 0(x ∈ F1); f(x) = 1/2(x ∈ F2); f(x) = 1(x ∈ F3). #�$ > f(x) = 1 2 [ d(x, F1) d(x, F1) + d(x, F2 ∪ F3) + d(x, F1 ∪ F2) d(x, F1 ∪ F2) + d(x, F3) ] . ( > š�$#£�¤�¥ ¦�§�¨�©�;�ª ™ ª�« Email: glhe@zjnu.cn, ¬ <�­�Z 2007.4.9 ) 8 ????? www.ucourse.net（??：??????：ucourse） ? 8 ?，? 19 ? 6/10/2009 ����� Lebesgue ����� �� �� Ex 1: � E ⊂ R1, ���� q ∈ (0, 1), ������������� (a, b), ����������ff {In}: E ∩ (a, b) ⊂ ∞⋃ n=1 In, ∞∑ n=1 m(In) < (b− a)q. fi�fl�ffi m(E) = 0. �! #"%$#&�'�( m∗(E ∩ (a, b)) ≤ ∞∑ n=1 m(In) < (b− a)q, )�*�+�,�-�.�/�0�1 I , 2 m∗(E ∩ I) ≤ q ∗ |I | 354 , E 65-5. L− 758 {In} 259 E ⊂ ∞⋃ n=1 In, :5;5<5=5>565?5@5A5@5B5C5259 m ∗(E) ≤ ∞∑ n=1 m∗(E ∩ In) ≤ q ∗ ∞∑ n=1 |In|, 3�4 2 m∗(E) ≤ qm∗(E) 3�4 m∗E = 0, D mE = 0. Ex 2: � A1, A2 ⊂ Rn, A1 ⊂ A2, A1 E�F�G�H �� m(A1) = m∗(A2) < ∞, fi�fl�ffi A2 E F�G�H�I �! #"%$#J A1 +�@�=�K�9ML Caratheodory &�'!N#O T = A2, 2 m∗(A2) = m ∗(A2 ∩ A1) + m ∗(A2 ∩A c 1) = m(A1) + m ∗(A2 \A1) $#J m(A1) = m ∗(A2) < ∞, 3�4 m∗(A2 \A1) = 0. ,�-�. T ⊂ Rn, P�?�Q�R A1 6�@�=�C�9MS�L Caratheodory &�'!N#O T ∩ A2 T�U�V�W K�9M2 m∗(T∩A2) = m ∗(T∩A2∩A1)+m ∗(T∩A2∩A c 1) = m ∗(T∩A1)+m ∗(T∩(A2\A1)) = m ∗(T∩A1). X�Y m∗T = m∗(T ∩A1) + m ∗(T ∩Ac1) = m ∗(T ∩ A2) + m ∗(T ∩Ac1) ≥ m∗(T ∩ A2) + m ∗(T ∩ Ac2) :�;[Z]\�6�^�_�` m∗T ≤ m∗(T ∩ A2) + m ∗(T ∩ Ac2) ( m∗T = m∗(T ∩ A2) + m ∗(T ∩ Ac2) a�b A2 +�@�=�K I c . "ed5f 65g5h5+5i5j5R Caratheodory &5' 9ek5l5m5n5i5j5R5@5=5K565C5o I $ m∗(A2\ A1) = 0, 3�4 A2 \A1 +�p�=�K�9M:�; A1 @�=�q�@�r�s A2 6�@�=�C I 1 ????? www.ucourse.net（??：??????：ucourse） ? 9 ?，? 19 ? 6/10/2009 Ex 3: � A, B ⊂ Rn � E�F�G�H 9 fi�fl�ffi m(A ∪ B) + m(A ∩ B) = m(A) + m(B). �t u"wv m(A), m(B) N 2yxyz U ∞, { Xy| �y} `y~y\yy€ I 3y45y‚ m(A) < ∞, m(B) < ∞. Q�R A, B 6�@�=�C�ƒ Caratheodory &�' 9M2 m(A ∪B) = m((A ∪ B) ∩ A) + m((A ∪ B) ∩ Ac) = m(A) + m(B \A), m(B) = m(B ∩A) + m(B \A), „ d�…�† `�‡�ˆ�S�‰�Ł ( c .�s m(A) < ∞, m(B) < ∞) q�r m(A ∪ B) + m(A ∩ B) = m(A) + m(B). Ex 4: fi�‹ E�Œ ���� H F ,F ⊂ [a, b] F 6= [a, b], Ž m(F ) = b− a? g "Mv F +��K�9 F ⊂ [a, b]  F 6= [a, b], { (a, b) \ F 6= ∅, q (a, b) \ F U x�z�‘�’�/ K�9M“�2 m((a, b) \ F ) > 0, ” $#J m([a, b]) = m(F ) + m([a, b] \ F ) 3�4 m(F ) = m([a, b])−m([a, b] \F ) ≤ m([a, b])−m((a, b) \F ) < b− a. D�•�–�— N#˜�™ 6� K F š�›�L I Ex 8: � {Ek} E [0, 1] œ� F�G�H ff�9 m(Ek) = 1 (k = 1, 2, · · ·), fi�fl�ffi m ( ∞⋂ k=1 Ek ) = 1. �ž Ÿ" $ m(Ek) = 1, { m(E c k) = 0( ¡5¢5£5K5^ J [0, 1] O ), 354 m( ∞⋃ k=1 Eck) ≤ ∞∑ k=1 m(Eck) = 0, a�b m( ∞⋂ k=1 Ek) = 1. Ex 9: � Ei (i = 1, 2, · · · , k) E [0, 1] œ� F�G�H 9M �� k∑ i=1 m(Ei) > k − 1, fi�fl�ffi m ( k⋂ i=1 Ei ) > 0. �ž Ÿ"¤$ k∑ i=1 m(Ei) > k−1 ( k− k∑ i=1 m(Ei) < 1, )5q k∑ i=1 (1−m(Ei)) < 1, q k∑ i=1 m(Eci ) < 1, { m ( k⋃ i=1 Eci ) ≤ k∑ i=1 m(Eci ) < 1 , 3�4 m ( k⋂ i=1 Ei ) > 0. ( ¡�¢�£�K�^ J [0, 1] O ). 2 ????? www.ucourse.net（??：??????：ucourse） ? 10 ?，? 19 ? 6/10/2009 Ex 14: fi�fl�ffi�¥ H E F�G �¦�§�¨�©�ª�« E " ����¬ ε > 0, ����� H G1, G2: G1 ⊇ E, G2 ⊇ E c, ��� m(G1 ∩G2) < ε. �! #"M&�' 6�“ ˜ C v E @�=�9#{�,�-�­�6 ε > 0, ›�L G1 +�/�K�9 E ⊂ G1,  m(G1 \E) < ε/2; ”�›�L F +��K�9 F ⊂ E,  m(E \ F ) < ε/2, O G2 = F c, { G2 +�/�K� G2 ⊇ E c,  m(G1 ∩G2) = m(G1 \ F ) ≤ m(G1 \E) + m(E \ F ) < ε. &�' 6�®�¯�C�9M¡�° &�' ‡�± J ,�-²­ ε > 0, ›�L G +�/�K�9M E ⊂ G,F +��K�9M F ⊂ E, b  m(E \ F ) < ε. m5n εk = 1/k, {5›5L5/50515³5A Gk ƒ550515³5A Fk ´ r59 Fk ⊂ E ⊂ Gk,m(Gk \Fk) < 1/k, µ G = ∞⋂ k=1 Gk, F = ∞⋃ k=1 Fk, { m(G \ F ) ≤ 1/k ,�-�. k ¶��€�9MD m(G \ F ) = 0, m�n s F ⊂ E ⊂ G, ( E @�= I Ex II(6): � A, B ⊂ Rn, A ∪ B E�F�G�H 9M m(A ∪ B) < ∞, · m(A ∪ B) = m∗(A) + m∗(B), fi�fl�ffi A, B ¸ F�G�I �! #" T B 6 } =�¹ H , q H ⊇ B, H @�=�9 $#J A ∪B @�=�9 3�4 š�º ‚ H ⊂ A ∪B, » { O H ∩ (A ∪ B) T�U B 6 } =�¹ I Q�R H 6�@�=�C�9 Y ƒ Caratheodory &�' 2 m(A ∪ B) = m∗((A ∪ B) ∩H) + m∗((A ∪ B) ∩Hc) = m∗(A \H) + m(H) = m∗(A \H) + m∗B c .�s m(A ∪B) = m∗(A) + m∗(B) Y ƒ m(A ∪ B) < ∞ 3�4 m∗(A \H) = m∗(A) Q�R H 6�@�=�C�9 Y ƒ Caratheodory &�' ”�2 m∗(A) = m∗(A \H) + m∗(A ∩H) X�Y m∗(A ∩H) = 0, a�b A ∩H +�@�=�K�9 $#J A = ((A ∪B) \H) ∪ (A ∩H) ( A @�= I B 6�@�=�C�¼�½ c .�s A, B ¾�¿�6�,� C�q�@ I ( T�À "MÁ�Â�à Ä�Å�Æ�Ç�È�É�Ê�É _ Email: glhe@zjnu.cn, Ë�Ì�Í�Î 2007.4.25) 3 ????? www.ucourse.net（??：??????：ucourse） ? 11 ?，? 19 ? 6/10/2009 ����� ������� �� ��� ������ Ex 1: ��������� I, {fα(x) : α ∈ I} � Rn ����ff�fi�fl�ffi��� "!�#�fl�ffi S(x) = sup{fα(x) : α ∈ I} $ Rn � � ff%fi%�%&(' )%*,+%-%.%/ I 0%1%2 /%3 S(x) 0%1%4%5%2%6,7%8 S(x) 9%:%;%1%4%6,<%=%:%>%9%1 4 / I , ? ∀α ∈ I @%5%2 fα(x) = { 1, x = α 0, A%B C%3 S(x) = sup{fα(x) : α ∈ I} = χI(x) 9%0%1%4%5%2%6 Ex 3: � f(x) $ [a, b)