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NAJDETSQ• OTKRYTAQ OKRESTNOSTX TO^KI a |
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KOTORAQ NE QWLQETSQ KORMU[KOJ SETI x. pOLOVIM |
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U(k) = Uk (aik ) iY6=ik Ei; k 2 f1; : : : ; sg; |
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I ZAMETIM, ^TO HOTQ BY ODIN \CILINDR" U(k) PRINADLEVIT |
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sLEDOWATELXNO, LOWU[KOJ QWLQETSQ I MNOVESTWO U = |
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U NE QWLQETSQ KORMU[KOJ DLQ x |
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TAK KAK ai1 | PREDELXNAQ TO^KA SETI pi1 (x), A S DRUGOJ STORONY U |
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x108. lOKALXNO KOMPAKTNYE PROSTRANSTWA
1. tOPOLOGI^ESKOE PROSTRANSTWO NAZYWAETSQ LOKALXNO KOMPAKTNYM, ESLI KAVDAQ EGO TO^KA OBLADAET KOMPAKTNOJ OKRESTNOSTX@. lOKALXNAQ KOMPAKTNOSTX | \TO TOPOLOGI^ESKOE SWOJSTWO. qSNO TAKVE, ^TO WSQKOE KOMPAKTNOE PROSTRANSTWO QWLQETSQ LOKALXNO KOMPAKTNYM.
p R I M E R Y. 2. Rn | LOKALXNO KOMPAKTNOE PROSTRANSTWO.
3.oTKRYTYJ [AR B1 ( ) W Rn | LOKALXNO KOMPAKTNOE PROSTRANSTWO.
4.dISKRETNOE PROSTRANSTWO LOKALXNO KOMPAKTNO.
pEREHODIM K IZU^ENI@ SWOJSTW LOKALXNO KOMPAKTNYH PROSTRANSTW.
5. wSQKOE OTDELIMOE LOKALXNO KOMPAKTNOE PROSTRANSTWO REGULQR-
NO.
pUSTX E | LOKALXNO KOMPAKTNOE PROSTRANSTWO, x 2 E I K | KOM- PAKTNAQ OKRESTNOSTX TO^KI x. tAK KAK K | REGULQRNOE PODPROSTRANSTWO E (105.8), SU]ESTWUET FUNDAMENTALXNAQ SISTEMA F ZAMKNUTYH OKREST- NOSTEJ TO^KI x W K ; TOGDA F | FUNDAMENTALXNAQ SISTEMA ZAMKNUTYH OKRESTNOSTEJ x W E: >
iSKL@^ITELXNO INTERESNYM FAKTOM QWLQETSQ TO OBSTOQTELXSTWO, ^TO DOBAWLENIEM ODNOJ (\BESKONE^NO UDALENNOJ• ") TO^KI LOKALXNO KOMPAKT- NOE PROSTRANSTWO MOVNO PREWRATITX W KOMPAKTNOE. pREDWARITELXNO RAS- SMOTRIM NESKOLXKO PRIMEROW.
p R I M E R Y. 6. sLEDU@]IE TRI LOKALXNO KOMPAKTNYH PROSTRANSTWA
POPARNO GOMEOMORFNY: R; (0; 1); S | OKRUVNOSTX BEZ ODNOJ TO^KI. eSLI
K S PRISOEDINITX WYBRO[ENNU@ TO^KU, TO POLU^IM OKRUVNOSTX S, KO- TORAQ QWLQETSQ KOMPAKTNYM PROSTRANSTWOM. tAKIM OBRAZOM, KAVDOE IZ TREH• PROSTRANSTW MOVNO \POGRUZITX" W KOMPAKTNOE PROSTRANSTWO, OTLI- ^A@]EESQ OT ISHODNOGO ODNOJ TO^KOJ.
7. lOKALXNO KOMPAKTNOE PROSTRANSTWO R2 POGRUVAETSQ W KOMPAKTNOE
2 |
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PROSTRANSTWO | EDINI^NU@ SFERU S( |
R3) S POMO]X@ STEREOGRAFI^ES- |
KOJ PROEKCII. R GOMEOMORFNO PRI \TOM S | SFERE S WYBRO[ENNOJ TO^- KOJ. iTAK, UKAZANNAQ KOMPAKTIFIKACIQ TAKVE OSU]ESTWLQETSQ PRISOEDI- NENIEM K R2 ODNOJ TO^KI.
8. t E O R E M A [p.s.aLEKSANDROW].dLQ KAVDOGO LOKALXNO KOMPAKTNO- GO PROSTRANSTWA E SU]ESTWUET KOMPAKTNOE PROSTRANSTWO E0 TAKOE, ^TO E GOMEOMORFNO NEKOTOROMU PODPROSTRANSTWU PROSTRANSTWA E0, DOPOLNENIE KOTOROGO (W E0) SWODITSQ K ODNOJ TO^KE.
pUSTX (E; |
T ) | LOKALXNO KOMPAKTNOE PROSTRANSTWO. pOLOVIM E0 = |
E [f!g, GDE f!g | ODNOTO^E^NOE MNOVESTWO. oPREDELIM TOPOLOGI@ T 0 W |
E0 : U |
2 T 0, ESLI U 2 T , LIBO U ESTX MNOVESTWO WIDA f!g[(EnK), GDE K |
| NEKOTORYJ KOMPAKT W TOPOLOGI^ESKOM PROSTRANSTWE E (PROWERXTE, ^TO |
T 0 NA SAMOM DELE ESTX TOPOLOGIQ). qSNO TAKVE, ^TO (E; T ) GOMEOMORF- |
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0 | TOPOLOGIQ, INDUCIROWANNAQ W |
E (KAK ^ASTI E0 |
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TOPOLOGIEJ T |
0. nAKONEC, (E0; T 0) | KOMPAKTNOE PROSTRANSTWO. dEJSTWI- |
TELXNO |
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PUSTX |
(U ) 2A | |
OTKRYTOE POKRYTIE |
E0. |
tOGDA NAJDETSQ |
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POLOGI^ESKOM PROSTRANSTWE (E; T ). pRI \TOM K0 |
2 |
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x109. sWQZNOSTX
TAKAQ, ^TO K0 S2 (U nf!g). oTS@DA (U ) 2 [f 0g | KONE^NOE POKRYTIE
E0: >
9. u P R A V N E N I E. pUSTX E | LOKALXNO KOMPAKTNOE PROSTRANSTWO,
E0 = E[f!g. oPREDELIM W E0 TOPOLOGI@ T 0 : U 2 T 0, ESLI0U 2 T , LIBO U ESTX MNOVESTWO WIDA f!g[V , GDE V 2 T . uBEDITESX, ^TO T | TOPOLOGIQ,
NO (E0; T 0) NE QWLQETSQ, WOOB]E GOWORQ, KOMPAKTNYM RAS[IRENIEM E.
1. tOPOLOGI^ESKOE PROSTRANSTWO (E; T ) NAZYWAETSQ SWQZNYM, ESLI NE SU]ESTWUET RAZBIENIQ E NA DWA NEPUSTYH OTKRYTYH MNOVESTWA, TO ESTX E NELXZQ PREDSTAWITX W WIDE E = U [ V , GDE U; V 2 T ; U 6= ;; V 6= ; I U \V = ;. ~ASTX X TOPOLOGI^ESKOGO PROSTRANSTWA E NAZYWAETSQ SWQZNOJ, ESLI W INDUCIROWANNOJ TOPOLOGII X | SWQZNOE PROSTRANSTWO. sWQZNOSTX QWLQETSQ TOPOLOGI^ESKIM SWOJSTWOM (!!).
p R I M E R Y. 2. oTREZOK [a; b] R SWQZEN f\TO SLEDUET, NAPRIMER, IZ LEMMY O WLOVENNYH OTREZKAHg.
3. eWKLIDOWO PROSTRANSTWO Rn SWQZNO.
oTMETIM TEPERX OSNOWNYE SWOJSTWA SWQZNYH MNOVESTW.
4. |
eSLI |
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SEMEJSTWO SWQZNYH ^ASTEJ TOPOLOGI^ESKOGO PRO |
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5. oBRAZ SWQZNOGO TOPOLOGI^ESKOGO PROSTRANSTWA PRI NEPRERYWNOM
OTOBRAVENII SWQZEN. |
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6. zAMYKANIE SWQZNOGO MNOVESTWA SWQZNO. |
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p.4. dOPUSTIM, NAPROTIW, ^TO Y = i |
I Xi |
NE SWQZNO. tOGDA SU]ESTWU@T |
U1; U2 2 T TAKIE, ^TO |
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EMYH PRAWOJ ^ASTI RAWENSTWA ( ). pUSTX, NAPRIMER, x 2 Y1 . pUSTX i0 2 I |
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p.5. pUSTX f : E ! E0 NEPRERYWNO I E SWQZNO. eSLI BY SU]ESTWO- WALO RAZBIENIE f(E) NA DWA OTKRYTYH MNOVESTWA U0; V 0, TO NEPUSTYE OTKRYTYE MNOVESTWA f,1(U0); f,1(V 0) RAZBIWALI BY E, ^TO NEWOZMOVNO.
p.6. pUSTX (E; T ) | TOPOLOGI^ESKOE PROSTRANSTWO I X( E) SWQZNO. pUSTX W TO VE WREMQ SU]ESTWU@T U1; U2 2 T TAKIE, ^TO
X, = (X, \ U1 ) [ (X, \ U2 ); X, \ Ui 6= ; (i = 1; 2);
(X, \ U1 ) \ (X, \ U2 ) = ;:
tOGDA MNOVESTWA X \ Ui (i = 1; 2) OBRAZU@T OTKRYTOE RAZBIENIE X: > u P R A V N E N I Q. 7. eSLI E; F | SWQZNYE TOPOLOGI^ESKIE PRO-
STRANSTWA, TO E F SWQZNO. |
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dOKAVITE ^TO |
GREBENKA |
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x110. lINEJNAQ SWQZNOSTX
1. tOPOLOGI^ESKOE PROSTRANSTWO E NAZYWAETSQ LINEJNO SWQZNYM, ES- LI L@BYE DWE EGO TO^KI x I y MOGUT BYTX SOEDINENY \PUT•EM", TO ESTX SU]ESTWUET NEPRERYWNOE OTOBRAVENIE f : [0; 1] ! E TAKOE, ^TO x = f(0), y = f (1). ~ASTX X TOPOLOGI^ESKOGO PROSTRANSTWA E NAZYWAETSQ LINEJNO SWQZNOJ, ESLI W INDUCIROWANNOJ TOPOLOGII X | LINEJNO SWQZNOE PRO- STRANSTWO. lINEJNAQ SWQZNOSTX QWLQETSQ TOPOLOGI^ESKIM SWOJSTWOM (!!).
2. lINEJNO SWQZNOE PROSTRANSTWO SWQZNO.
pUSTX x0 | PROIZWOLXNAQ TO^KA W LINEJNO SWQZNOM PROSTRANSTWE E |
I DLQ KAVDOGO x 2 E fx : [0; 1] ! E | PUTX, SOEDINQ@]IJ TO^KU x0 |
S TO^KOJ x. uTWERVDENIE SLEDUET IZ PREDSTAWLENIQ E = |
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U^•ETOM 109.4. |
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3. z A M E ^ A N I E. iZ SWQZNOSTI LINEJNAQ SWQZNOSTX NE SLEDUET. nAPRIMER, \GREB•ENKA" (SM. 109.8) | SWQZNOE, NO NE LINEJNO SWQZNOE PRO- STRANSTWO (!!).
u P R A V N E N I Q. 4. sWOJSTWA 109.4{5 OSTA@TSQ SPRAWEDLIWYMI DLQ LINEJNO SWQZNYH MNOVESTW.
5.wSQKOE SWQZNOE PODMNOVESTWO MNOVESTWA RQWLQETSQ I LINEJNO SWQZ- NYM. oPI[ITE WSE SWQZNYE PODMNOVESTWA R.
6.iSSLEDOWATX NA SWQZNOSTX I LINEJNU@ SWQZNOSTX SLEDU@]IE MNO- VESTWA:
(A) Q (W R ), (B) RnQ (W R), (W) Q2 (W R2).
(G) = f(x; y) 2 R2j x 2 Q LIBO y 2 Qg ( W R2).
(D) = f(x; y) 2 R2j \x 2 Q; y 62Q" LIBO \x 62Q; y 2 Q"g,
7. wSQKOE WYPUKLOE MNOVESTWO W Rn LINEJNO SWQZNO.
mera vordana
oB]AQ IDEQ MEROOPREDELENIQ ZAKL@^AETSQ W PRODOLVENII MERY S \\LE- MENTARNYH" MNOVESTW, GDE MERA UVE OPREDELENA NEKOTORYM ESTESTWEN- NYM OBRAZOM, NA BOLEE [IROKIJ KLASS \IZMERIMYH" MNOVESTW S SOHRA- NENIEM SWOIH OSNOWNYH SWOJSTW (NEOTRICATELXNOSTX I ADDITIWNOSTX). nIVE BUDET IZLOVENO POSTROENIE MERY PO vORDANU W EWKLIDOWOM PRO- STRANSTWE Rn. iZLOVENIE SNA^ALA BUDET WESTISX DLQ SLU^AQ R2 TOLXKO DLQ BOLX[EJ NAGLQDNOSTI.
x111. |LEMENTARNYE MNOVESTWA
1. pRQMOUGOLXNIKOM W R2 (SO STORONAMI, PARALLELXNYMI OSQM KOOR-
DINAT) NAZYWAETSQ MNOVESTWO WIDA
= f(x; y) 2 R2j x 2 ha; bi; y 2 hc; dig = ha; bi hc; di;
GDE ^EREZ ha; bi OBOZNA^AETSQ ODIN IZ PROMEVUTKOW WIDA (a; b), [a; b); (a; b], [a; b] (a; b 2 R). mNOVESTWO E( R2) NAZYWAETSQ \LEMENTARNYM, ESLI ONO QWLQETSQ OB_EDINENIEM KONE^NOGO ^ISLA POPARNO NEPERESEKA@]IHSQ PRQ- MOUGOLXNIKOW:
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POPARNO NEPERESEKA@]IHSQ MNOVESTWAH. tAKIM OBRAZOM, WMESTO ( ) BU- |
DEM PISATX E = 1 |
+ : : : + n |
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i. |
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oBOZNA^IM ^EREZ E KLASS WSEH \LEMENTARNYH MNOVESTW W R2 . oTMETIM WAVNYE DLQ NAS SWOJSTWA \TOGO KLASSA.
pUSTX E I F | PROIZWOLXNYE MNOVESTWA IZ KLASSA E. tOGDA
2.E [ F; E \ F 2 E ,
3.ESLI E F , TO FnE 2 E,
4.SU]ESTWU@T POPARNO NEPERESEKA@]IESQ PRQMOUGOLXNIKI 1; : : : ; n
TAKIE, ^TO E = P i; F = P i, GDE ; 0 f1; : : : ; ng.
i2 i2 0
sNA^ALA USTANOWIM PROSTU@ GEOMETRI^ESKU@ LEMMU.
5. eSLI PRQMOUGOLXNIKI 1; : : : ; k POPARNO NE PERESEKA@TSQ I WSE SODERVATSQ W PRQMOUGOLXNIKE , TO SU]ESTWU@T POPARNO NEPERESEKA-
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@]IESQ PRQMOUGOLXNIKI k+1; : : : ; n TAKIE, ^TO = |
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dOKAZATELXSTWO PROWODITSQ INDUKCIEJ PO k. dLQ k = 1 SPRAWEDLIWOSTX UTWERVDENIQ USTANAWLIWAETSQ PEREBOROM WOZMOVNYH SLU^AEW RASPOLOVE- NIQ PRQMOUGOLXNIKOW. (pRIMER ODNOGO IZ WOZMOVNYH SLU^AEW PRIWED<N
NA rIS. 20.) |
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dOPUSTIM, ^TO UTWERVDENIE WERNO |
DLQ |
WSEH |
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NATURALXNYH |
^ISEL |
k , 1; |
I PUSTX SEMEJSTWO |
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1; : : : ; k |
UDOWLETWORQET USLOWIQM P |
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pO |
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i + j |
Pj , GDE fPj g |
PREDPOLOVENI@, IMEET MESTO PREDSTAWLENIE = i=1 |
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| NEKOTORAQ KONE^NAQ SISTEMA PRQMOUGOLXNIKOW. pOLOVIM 0 |
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tOGDA 0 |
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Pj I, SLEDOWATELXNO, Pj = 0 |
+ (j) , GDE |
(j) | PRQMOUGOLX- |
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NIKI I s PROBEGAET KONE^NOE MNOVESTWO INDEKSOWP |
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6. pEREHODIM K DOKAZATELXSTWU PP. 2{4. uSTANOWIM SNA^ALA P. 4. pUSTX |
E = n i; F = |
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s 0 . pOLOVIM ij = i |
0 ; \TI PRQMOUGOLXNIKI |
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POPARNO NE PERESEKA@TSQ. tAK KAK ij 0j; ij i, TO W SILU P. 5: |
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tOGDA SEMEJSTWO |
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pP. 2, 3 QWLQ@TSQ SLEDSTWIQMI P. 4. dEJSTWITELXNO, W OBOZNA^ENIQH
P. 4 IMEEM
2Xn |
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EnF = |
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i; E [ F = |
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x112. mERA NA KLASSE \LEMENTARNYH MNOVESTW
1. mEROJ (PLO]ADX@) PRQMOUGOLXNIKA = ha; bi hc; di NAZYWAETSQ ^ISLO m( ) = (b,a)(d, c). w ^ASTNOSTI, ESLI WYROVDEN (TO ESTX a = b ILI c = d), TO m( ) = 0: sLEDU@]EE WAVNOE SWOJSTWO MERY NA KLASSE PRQMOUGOLXNIKOW NAZYWAETSQ ADDITIWNOSTX@:
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2. eSLI = |
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k, |
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m( k ). |
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pUSTX (a = x0 < x1 < : : : < xn = b); 0(c = y0 |
< : : : < ys = d) | |
RAZLOVENIQ OTREZKOW [a; b] I [c; d]. eSLI = |
i;j |
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1 |
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TO PREDSTAWLENIE P |
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yj,1; yj (i = 1; n; j = 1; s), |
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NAZOWEM REGULQR |
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NYM. nETRUDNO WIDETX, ^TO UTWERVDENIE WERNO DLQ SLU^AQ REGULQRNOGO |
PREDSTAWLENIQ: |
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m( ) = (b a)(d c) = [ |
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oB]IJ SLU^AJ LEGKO SWODITSQ K REGULQRNOMU (!!). |
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3. pRODOLVIM MERU NA KLASS E WSEH \LEMENTARNYH MNOVESTW: DLQ E = |
n |
i POLOVIM m(E) = |
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uBEDIMSQ W KORREKTNOSTI DANNOGO OPREDELENIQ. pUSTX E = |
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j=1 |
| E]E• ODNO PREDSTAWLENIE E W WIDE OB_EDINENIQ POPARNO NEPERESEKA- |
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m( 0 ). dEJSTWITELX- |
@]IHSQ PRQMOUGOLXNIKOW. tOGDA |
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m( i) |
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NO, POLAGAQ ij = i |
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0 |
, ZAMETIM, ^TO SPRAWEDLIWY RAWENSTWA E = |
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P P |
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ij. pO\TOMU m( i) = |
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m( ij ); m( 0 ) = |
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m( ij), OTKUDA |
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X |
m( i) = |
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m( ij ) = |
X X |
m( ij ) = |
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m( 0 |
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i=1 j=1 |
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j=1 |
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iZ DANNOGO OPREDELENIQ SRAZU SLEDUET, ^TO MERA m NA KLASSE E PO- PREVNEMU OBLADAET SWOJSTWOM ADDITIWNOSTI:
4. eSLI E; F 2 E I E \ F = ;, TO m(E + F ) = m(E) + m(F ). oTMETIM E]E• NESKOLXKO POLEZNYH SWOJSTW MERY (!!):
5.eSLI E 2 E, TO E ; E, 2 E I m(E) = m(E ) = m(E,).
6.eSLI E F (E; F 2 E), TO m(F ) = m(E) + m(F nE).
7.eSLI E; F 2 E, TO m(E [ F ) m(E ) + m(F ).
8.eSLI = , A PRQMOUGOLXNIK 1 NE WYROVDEN I \ 1 6= ;, TO m( \ 1) > 0.
x113. sWOJSTWO S^ETNOJ• ADDITIWNOSTI
mERA NA KLASSE PRQMOUGOLXNIKOW OBLADAET SWOJSTWOM SU]ESTWENNO BOLEE SILXNYM, ^EM SWOJSTWO 112.2. oNO NAZYWAETSQ S^•ETNOJ ADDITIW- NOSTX@ I LEVIT W OSNOWE PRINCIPIALXNO NOWOJ TEORII MERY I INTEG- RALA, KOTORAQ IZLAGAETSQ NIVE, W RAZDELAH \mERA lEBEGA" I \iNTEGRAL lEBEGA".
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1. eSLI = 1 |
k, GDE ; k |
| PRQMOUGOLXNIKI, TO |
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k=1 |
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m( ) = |
1 m( k): |
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k=1 |
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2 n |
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zAFIKSIRUEM PROIZWOLXNOE n |
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N. w SILU 111.5 NAJDUTSQ PRQMOUGOLX- |
NIKI 0 |
; : : : ; 0 TAKIE, ^TO = |
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k + |
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0 , I W SILU ADDITIWNOSTI |
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n+1 |
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k=1 |
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j=n+1 |
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I NEOTRICATELXNOSTI MERY m: |
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m( k) |
k=1 |
m( k ) + |
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m( 0 ) = m( k + |
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0 ) = m( ): |
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k=1 |
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j=n+1 |
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iZ PROIZWOLXNOSTI n TEPERX POLU^AEM |
1 m( k) |
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m( ). oBRATNOE NE- |
RAWENSTWO SLEDUET IZ UTWERVDENIQ: |
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kP=1 |
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1 |
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1 |
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2. eSLI k=1 |
k, TO m( ) |
k=1 m( k ). |
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pUSTX " > 0 PROIZWOLXNO I | ZAMKNUTYJ PRQMOUGOLXNIK TAKOJ, |
^TO I m( ) |
m( ) + "=2. dLQ KAVDOGO k RASSMOTRIM OTKRYTYJ |
PRQMOUGOLXNIK k TAKOJ, ^TO |
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(k+1) |
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" (k = 1; 2; : : :): |
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; m( k) < m( k) + 2, |
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179 |
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1 |
k . sEMEJSTWO |
1; 2; : : : OBRAZUET OTKRYTOE POKRY- |
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qSNO, ^TO k=1 |
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f f |
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. |
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, |
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SU]ESTWUET KONE^NOE SE |
- |
TIE KOMPAKTNOGO MNOVESTWA |
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sLEDOWATELXNO |
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MEJSTWO 1 |
; : : : N, KOTOROE POKRYWAET : |
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k=1 |
k . w SILU 112.7 |
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m( ) k=1 m( k). sLEDOWATELXNO, |
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1 |
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" |
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m( ) m( ) + 2 |
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X |
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m( k) + ": |
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k=1 |
m( k ) + 2 |
k=1 |
m( k ) + 2 |
k=1 |
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iZ PROIZWOLXNOSTI " P. 2 DOKAZAN, A WMESTE S NIM I P. 1. |
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> |
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x114. iZMERIMYE PO vORDANU MNOVESTWA
1. dLQ OGRANI^ENNOGO MNOVESTWA X( Rn) OPREDELENY DWA ^ISLA:
m (X ) supfm(E)j E X; E 2 Eg | WNUTRENNQQ MERA vORDANA MNOVESTWA X,
m (X ) inffm(E)j X E; E 2 Eg | WNE[NQQ MERA vORDANA MNO- VESTWA X.
oTMETIM, ^TO m (X) m (X ) DLQ PROIZWOLXNOGO OGRANI^ENNOGO MNO- VESTWA X . oGRANI^ENNOE MNOVESTWO X( Rn ) NAZYWAETSQ IZMERIMYM PO vORDANU (J-IZMERIMYM) , ESLI m (X) = m (X). ~ISLO m(X)
WAm (XX.) (= m (X )) NAZYWAETSQ MEROJ vORDANA J -IZMERIMOGO MNOVEST-
n
z A M E ^ A N I Q. 2. kAVDOE \LEMENTARNOE MNOVESTWO E = P k
n
k=1
J -IZMERIMO I EGO MERA vORDANA RAWNA P m( k ). zDESX I DALEE BUKWOJ
k=1
OBOZNA^A@TSQ (n-MERNYE) PARALLELEPIPEDY ha1; b1i : : : han; bni.
3. iZ OPREDELENIQ P. 1 MNOVESTWO X IMEET VORDANOWU MERU NULX
(m(X ) = 0) TTOGDA 8" > 0 9E 2 E (X E; m(E) < "). oTMETIM SWOJSTWA MNOVESTW VORDANOWOJ MERY NULX:
4.m(X) = m(Y ) = 0 ) m(X [ Y ) = 0,
5.Y X; m(X ) = 0 ) Y J-IZMERIMO I m(Y ) = 0.