ZE SZYT Y NAUKOWE WYŻ SZEJ SZKOŁY PEDAGOGICZNEJ W BYDGOSZCZY P r o b le m y M a te m a ty cz n e 1982 z . 3
WŁODZIMIERZ ŚLĘZAK B y d g o s z c z
H3MEPHM0CTL ffirHKĘHH HECKOJILKPIX IEPEMEHHHX
B
3Tot paóoTe imojEeHa HeKOTopan MOflHguncamw TeopeMH H3
c t e t b h A /. nojiyaeHa nyreM BBeaeHnn noaaTM t - zjm
nereHe-
pamw
BMecro ^ereHepamui.Mbi acnojn>3yeM cjieflyrajyio nem y:
JleMMa I . / U j t Jiema
2 ,C T p .II4 / . IlycTB flc^óyneT H3Mepn-
mhm no HeCery MHoaectbołi. Torna cymecTByeT nojęMHOsectbo BcrA,
Tose H
3MepHMoe .TaKoe.trro mJ.A~&)~0 /rn e vnK- Mepa Jledera b £ / '
h KpoMe Toro B c - + B aro no onpeaejieHmo
3Hawr,nT0 Kanaan
t o h - Ka ( x „ X Ł,../x nje B ya oB J ieTB op a eT c jie n y m w M 3-m ycjiOBHHM: ( i ) ( x „ x „ . . . / „ )HBnneTCH TOnKOfi IIJIOTHOCTI
1 B ^ J ^ HBJIfleTCfl TOnKOH iuiothocth
MHOSeCTBa
:
vx")e&][
WIR Bcex
i i
W .
0 > ? (x Ł ~ .,XV)j e ? >ł* 1flBJiaeTCH t o e k oK h j io t h o c t h cen eH u n
BoT*ocHOBHa
rTeopeMa:
T e op e M a I . Ily c T Ł ó y n e T neficTB H TejiB H oii m y m a w e ii T a
-k o 2 , m to B ce e e c en eH n n t * — * 4-» >. y . = 1xa) - j > * l + 4 ) " y ' '
—
66
—i - 1r ~j* K3MepnMHe, a Bce ceaeHUH
OdJianaDT CBOiłCTBOM
/cm.
/ Z / /.®yHKOHfl HBJWeTCH npH 9THX
npejyioseHHHX H3Mepumom Torna
a tojibko’ Torna,Korna MHOsecTBO
( { ) = . U f ( X n X D - ' V X n ) •’ *
HBJifleTcs
b TOHKe X ;nonosHTejiBHo
| - 3nereHepoBaHa
hjih3a e r e H e p o B a H a j - HBJiaeTCfl H3MepnMUM ,M ep u H yjiB .
£ 0 Ka3aTejiBCTB0 a/HeoóxoBKM OCTB. KoHe*jHO
f\( f i C U { ( x 4l * l ) " - J X* ) € ^
•; *i-«i)X;4 t/"\>
1
.
He
HBJweTCH annpoKCHMaTHBHO HenpepuBHa b tomto X ;j,i\ n e
cyM-MHpoBaHne óep eT C H no MOHy.ro
>i
,M H oxecTBOc
npaBOft CTopoHH3TOrO BKTOHeHHH H3Mep0MO,H K8 K n0 K a 38H0 B
/Z/
C T p . I I 5 - I I 6 3 T0 n o cym e cT B y Hyjn> - M HOsecTBO.OoaTOMy nojiy^HM T o s e ,m nW i p o
<5/flocT aT O *3HOCTB /orpaHHHHMaH c jiya a eM n. = 3/ .Ilo K a x e M k t o y n o-BJieTBopneT ycnoBHHM jieMMH J le to H c a -r p a H H a r o / / j / / , K O Topaa h
c-n o jiB 30B ajiacB
yxe
b/57 .
IlyCTB E T c R ^ y n e T H3MepZMHM MHOseCTBOM IIOJIOMTeJIBHOft MepH
,
W j(hJ> 0 H
n y cT Bt > 0 -
3an aH floe r o ó o e n ojiO H krejiB H oe h h c jio .Oc
5—
p a 3yeM n o c jien o B a T ejiB H o cT B ( ^ k ) B c e x o t k p h t h x H HTepBanoB, k o t o -
p u x k oh iih pam ioHajiBHH. 0 ó p a 3yeM T o s e nocjrenoBaTejiBHOCTB ( K k )
3aMKHyTux j cerM eH TO B, kotopkdc h jih hh ynoBJieTBopfflDT H ep aB eH cT B y
i
H KOTOpHX KOHIIH pailHOHaJIBHH.-
6 7
-OÓ03HaHHM a e p e 3 Q r j MHOKeCTBO [ ( x Ł , X j ) 6 R.1 '- E "X^ )X n
K3MepHM0e MHOKeCTBO nOJIOEHTejIBHOil MepH, Z KpOMe T o r o *j* C"ł)
n pH H ajw esH T M H osecT B y K s npir b c h k o h t o t o n p iiH aiy ieaam eg ■
d - BHyTpeHHOCTH M H O S e C T B a n F A jJ ^ T O 3HaHHT i - TOHKa IIJIOTHO- CTH 3TOrO MHOSeCTBa/. IlyCTB
H3MepHM0, nCUIOMTejIBHOfi Mepu j .
noCKOJIBKy BCe Ce^eHHH | x T,X j yHOBJieTBOpHBT yCJIOBKB ( G ) (cm . I l i ) ,
TO y U 0 v p Q . T a K KaK KpoMe T o r o Q H B ^ a e T c a mhos6CTbom n o jr o -
SHTejIBHOft MepH, TO CymeCTBy6T COdOKyPHOCTB HHfleKCOB ('Co , Sc)
T a K a a , h t o ^ [ Q Lx.s^ > 0 , r ^ e m * BHemafiH n jio c K a a J le ó e r o B a M epa .
0óO3HaqnM q e p e 3 b MHOKecTBO j ( X i ) X j ) € R1 : ( x z , X 3) HBJweTCH
t o h k o 2 BHemHeK i m o t h o c t h MHoacecTBa Q y 0s ^ . MHosecTBO b H3M epn-
mo z KpoMe T o r o Wl Ł (B )= v n * (Q v os j)> 0 • ItojioKHM T en ep B c - M J n E .
O t o b m h o MHOKecTBO C H3MepHMO z e r o M epa noJioacHTejiBHaji, n o c -
KOJiBKy /w i1( C Xi)xi )n ojroK H Te^B H a m h o o h t h B c e x T o n eK
( x Ł ) x i ) e Q v . i
„
ń y cT B D = C n ( R * — R ( f ) ) . Ha ocHOBaHHH jieMMH I cym ecT B y eT
M S M H osecTB a D T a n o e MHosecTBO Z c D , h t o ryi3 ( D - 2 ) = 0 z KpoMe
T o r o Z c - + Z . O neB M H o Z BwncraeHo b E z e r o M epa / b T p o t e p n o M
n p ocTpaH C TB e/ nojiO K KTejiLH afi.
fV *
t o h k s U 4 , X3) npjJH ew iesam nq M H osecTB y Z . 3a$HKCnpyeM ( x , , x 3; x 3) e £ .
T a n KaK Z T H s ejio BKjno^aeTCH b c e (3a ( Z c - Z ) « T0 Z a a r a e T c a
( cl - OTKpHTHM, H3MepHMHM MHOSeCTBOM IIOjlOKHTeJIBHOji MepU. KpOMe
Toro KaS^HK H3 e r o H3MepHMHX HOflMHOSeCTB nOJIOHHTejIBHOii MepH
n e p e c e K a e T c a
c Qr , s, .
IIpeOToaosHM /h t o ó h n o jiya u T B n p o T H B o p e a iie / , h t o |>( x , ) ^ ,x3)
He npH H ajyiesH T . fcyHKOHH
( - U i j U j l - » l * * j ) Ha OCHOBaHHH TeopeM H3
Jsj
HBJIHeTCH H3MepHMOfi Ha MHOSeCTBeZ x
. HoCKOJIBKy KpOMe T o r o | x , C Q r . s # ) c . K S e;m e e M ( * ] * n 2 ^ K s j j ' 0-C a p y ro ft cTopoHH raeeM h ceKOHH
i v b TOHKe X, He HBmieTCfl hh f - hh J - 3,nereHepoBaHHoa;
s t o o s H a a a e T , h t o a * * ( Z x „ x 3 0 { v f } U
U 2 * , , *3 : { ( x „ x t tX j ) > l > l c } ) > 0 .
t o ł K a s flo a T0HKH 'Uj, £ Z Xi ( x 3 f ) ( x ^ j CeKOHH
b TOHKe He oóJiaaaB T c b oMc t b o m ^ - n e r e H e p a iin n h h
j , - a e r e H e p a m m noaTOM y /vn,2 * ( Z ^ n C l ^ ) 1 O
n p o T itB o p eh h t [%-J h 3aB epm aeT fl0 K a 3 a T ejiŁ C T B 0 .
C j i e H C T B H e : D y c T B •' d y z e r aeficTB H T ejiB H oa
^byHKiaiea o t 3 nepeM eHm oc x 1) x l ) X i . E cjih BCe e e ceneHHH | x „ , Xj )
■fxl ( x 3 ) f x 1,xJ h b jih d t c h npoH 3BOBHHMH, t o npH H ajyiesH T II] K x a c -
c y E e p a . B £ 0 K a3aTejiBC TBe cymecTBeHHym p o jib n r p a e T H3M e p n
--
6 9
-m o c t ł ftyHK-mra k o to pan BHTeitasT
03
HameB TeopeMH I .B
OCHOBe pe3yjIŁTaTOB H3JI0EeHHHX B 3TOH CTaTBe n e s a Trn y -
ÓOKae w e n 3 . r p a ą n 3 , KOTopoMy aBTop oqeHB óJiaroaapeH 3 a h h c- nupamno h noMoniB. J lH T epaTypa / 1 / Z .G r a n d e , l a m e s u r a b i l i t e d e s f o n c t i o n s d e d e u x v a r i a b l e s e t d e l a s u p e r p o s i t i o n F( x , f ( x )) , D i s s e r t a t i o n e s ŁMa th e m a - t i c a e , CL I X / 1 9 7 8 / , p . 1 - 5 0 . / 2 / - S u r l a m e s u r a b i l i t e d e s f o n c t i o n s d e p l u s i e u r s v a r i a b l e s , Ma t h . S l o v a c a 28 / 1 9 7 8 / , No . r , p .1 1 3 - 1 1 8 / 3 / A . M a r c z e w s k i, C z . R y l l - Na r d z e w s k i , S u r l a m e s u r a b i l i t e d e s f o n c t i o n s d e p l u s i e u r s v a r i a b l e s , A n n . S o c . P o l . Ma t h . 25 / 1 9 5 2 / , p . 145-154 / 4 / Z . Z a h o r s k i , S u r l a p r e m i ś r e d e r i v e e , T r a n s .A m e r . Ma t h . S o c . 69 / 1 9 5 0 / , p .1 - 5 4 / 5 / W . Ś l ę z a k , Une c o n d i t i o n e q u i v a l e n t e a l a me s u r a b i l i t e d 'u n e f o n c t i o n d e d e u x v a r i a b l e s , / s u b m it e d t o P r o b le m y Ma t e m a t y c z n e 3 /
K
3 MEPKM0CTB &7 HKIIHH HECKOJIŁKffi IIEPEMEHHHX
CoaepataHHe
B p a ó o T e oP o d m a e T c fl HeKOTopne TeopeM H 3 .r p a H £ 3 . H aiiLO jiee
HHTepeCHHM CJiejICTBOM HBJIHeTCH TO , MTO $yHKU0H 3 n ep eM eH H ia ,
KOTopoB B ce c e ia ia K h b jih b t c h iip o h 3bohhhmh, npuH aiuiesH T 3 -M y
