Problemy Matematyczne 14 (1995), 21 - 25
On the sup-measurability of multifunctions
whose values are allowed to be noncompact
Grażjma Kwiecińska
W łodzimierz Ślęzak:
T h e content o f the work was announced during the Summer School on Real Functions Theory 1992. It concerns sorne generalizations o f Zygm unt’s theorem on the sup-measurabilitj' of multifunctions with the Caratheodory condition. T h e theorems are given without proofs, which can be found in [5].
Let
( X,
A 4 (A ')) be a measurable space, (K, d) a complete separable m etric space and( Z, g)
a m etric space. Suppose thatX )
has the following„p rojection p roperty” :
I f A € M ( X ) ® B o ( Y ) then ttx( A ) = {a; : 3y € Y ( x , y ) _ £ A } £ M ( X )
where B o ( Y ) denotes the cr-field of Borel sets in Y and Ad ( A ) denotes a com pletion o f M . { X ) .
A m ultifunction F : X x Y —* Z is M ( X ) sup-measurable if for each •M (A ')m easu rable multifunction G : X —> Y with closed values the superpo- sition $ : X —*
Z
defined by formuła $ ( x ) = F ( x , G ( x ) ) = Uy&a ( x ) f ' ( x , y )is A d (A r)measurable, i.e. $ + (i? ) = {x 6 X : $ ( x ) C B } G M . ( X ) for each open set B C Z .
T h e o re m 1 Let
( Z, g)
be acompact. Suppose that F :X
x Y —>Z
is a multifunction with A4(A)m easurable all y-sections and m ightly lower semicontinous all x-sections. Then F is A4(A)sup-m easurable.
T h e m ightly lower semicontinuity of multifunction: H : Y —►
Z
means that it is lower semicontinuous and for each point y £ Y there exists an open22 O N T H E S U P - M E A S U R A B I L I T Y OF M U L T I F U N C T I O N S
set V ( y ) C Y such that y € C l ( V ( y ) ) and H |{y}ui'(y) is upper semicontinuous at y.
Let we observe that the lower semicontinuity of all y-sections o f F is not sufficient to the Ad(A')sup-mea.surability of F . Consider for exam ple a multifunction F : R x R —> R defined by formuła
F { x , y ) =
{0 , 1 } if x ^ y
{1 } if x = y and x 0 4 {0 } if x = y and x (£ A
where A C R and A (fc C ( R ) , i.e. A is a nonmeasurable (in the Lebesgue sense) set. A ll x-sections of F are lower semicontinuos but not m ightly lower semicontinuous and al y-sections are £(i?)m easurable. If G ( x ) = { x } we have <h(x) = F ( x , { x } ) = { \ ^ ( x ) } . $ is in this case the characteristic function of the nonmeasurable set.
T h e o r e m 2 Let Z, o be cr-compact. Let T { Y ) be a topology in
Y
finer than the metric one such that ( Y , T ( Y ) ) is separable. Let us fix some countable dense subset of Y and denote it by S. Suppose that for every point v €
Y
there exist a subset U ( v ) 6T ( Y )
such that for each y € SB y = {u €
Y
: y € U ( v ) } € Bo( Y, d) and for everyv in Y
the fam ilyAf {Y)
= { ( 7 ( r ) fi K ( v , 2 ~ n) : n 6 Ar} forms a filter-ba.se of T (F )-n eigh b o u rgh oo d s of point v. Assume thatF
:X
xY
—> Z is a closed-valued multifunction whose all y-sections are Admeasurable and all x-sections are T (i')-co n tin u o u s. ThenF is Ad(A')sup-measurable.
E x a m p le Let ( Y , d , < ) be a partially ordered m etric space such that
( Y, < ) is a partially ordered set and there is a countable dense set S in (K, d)
such that for any y € Y we ha.ve: y = lim n_oo yn, for some seąuence yn € 5 and yn > y for n £ N . Let c a l T { Y ) be a topology on Y generated by all open sets in ( Y , d ) and also by all the intervals I a = {y € Y : y < a ] for a € Y. This topology fulfils the assumptions o f the theorem 2 as it have shown by Dravecky and Neubrunn (see theorem 6.5 on p. 156 in [2]). Fromtheorem 2 we obtain a corollary.
C o r o lla r y A multifunction F : X x R —+ R having all x-sections right- continuous (left-continuous) and all y-sections A ^ A ^ m ea su ra b le is Ad(A^)sup- -measurable, provided that values of F are closed.
R e m a rk Let A C R and A £ C ( R ) . Let F : R x R —* R be given by formuła
On T H E S U P - M E A S U R A B I L I T Y OF M U L T I F U N C T I O N S 23
f (1.2) if x € A and y < x
F ( x . y ) = < (1.2) if x (£A and y < x ( {0 } in other cases.
Some o f its x-sections are right-continuous, remaining are left-continuous, all y-sections are £(/?)measurable but it is not £(/?)sup-measurable.
This exam ple shows that the assumption of one-side continuity from the same side for all x-sections is essential.
T h e o r e m 3 Let ( A , T ( A ) ) be a perfectly normal topological space and let (Z , o) be cr-compact. If F : A x V —> Z is a multifunction with T ( A )-lower semicontinuous y-section and upper semicontinuous x-sections, then F is Z?o(AOsup-measurable.
R e m a rk 2 Let T ( X ) be the density topology in A . A multifunction
H : A —> Z is called approximately lower semicontinuous if H ~ ( U ) = { x €
A ’ : H ( x ) D U ^ 0 } € T (A " ) for each open set U C Z . In this case theorem 3 is a generalization Grande's theorem onto the case of multifunctions (see [4], theorem 30).
Let (Y, d, A A { Y ) , y) be a complete separable metric space with cr-finite regular com plete measure defined on a cr-field A i ( Y ) of subset of V containing £ o (T ). Let V ( Y ) be a differentition basis for the space ( Y, d, A 4 (V ), y ) (see [lj. p. 30) fulfilling the following conditions:
1. T>( Y) C M { Y ) is a countable fam ily of sets with nonempty interiors and positive frnite measure /z
2. For everv point y
€
Y there exists a seąuence of sets(/n)^Li
from V { Y ) converging to y, i.e. y 6 I n t ( I n) for n € N and seąuence of diameters of I n converges to zero when n approaches innfinity.( * ). A set A C Y has the property ( Z ) with respect to T>[ Y) if for any point
y € Y there exist an open set U C A and a number 8 > 0 such that n ( J n U ) 1
— > -n ( j ) 2
for each set J € P ( V ’ ) containing y and with diameter less than 6.
Denote by Z the fam ily of subsets of Y with the property ( Z ) . A m ulti
function H : Y —> Z has the property { Z ) if H ~ { G ) € Z and H + ( G ) € Z for each open set G C Z .
24 On T H E S E P - M E A S U RA BI L I T Y OF M U L T I F U N C T I O N S
T h e o r e m 4 Let (A L Z 3 (A )) be the Baire space, ( Y . d , M ( Y ) , f i ) let be as in ( * ) and let Z be cr-compact. Suppose that F : X x A —> Z is a mul tifunction such that all its x-sections ha.ve the property ( Z ) with respect to
c a l D ( Y ) and all its y-functions have the Baire property. Then F is Z3(Ar)sup-
measurable, where 5 ( A’ ) denotes the crfield of the subsets o f X with Baire property.
Theorem 4 is a generalization of theorem E. Grandę and Z. Grandę (see [3], theorem 1) onto the case of multifunctions.
Each of these above theorems serve as a kind of generalization o f Zyg- m unt’s theorem (see [7], theorem 2):
T h e o r e m (Z y g m u n t) If all x-sections of multifunction F : X x Y —> Z with compact values are continuous and all its y-sections are A f ( A ’ )sup-mea- surable, then F is A4(A')sup-measura.ble.
References
[1] Bruckner A .M ., Differentiation o f integrals, Am er. Math. M onthly 78 (1979), 1 - 54.
[2] Dravecky J., Neubrunn T ., Measurability o f functions o f two variables, M atem aticky Ćasopis 23 no 2, (1973), 147 - 157.
[3] Grandę E., Grandę Z., Quelqu.es remaręues sur la superposition
F ( x , f ( x ) ) , Fund. Math. C X X I (1984), 199 - 211.
[4] Grandę Z., La mesurabilite des fonctions de deux uariables et de la su
perposition F ( x , f ( x ) ) , Dissertationes mathematicae C L IX , 1 - 49.
[5] Kwiecińska G.,Ślęzak W ., Sup-mtasurability o f m ultifunctions, to appear. [6] Spakowski A., On Superpositionally Measurable M ultifunctions, A cta
Univ. Carolinae, vol. 30 no 2 (1989), 149 - 151.
[7] Zygmunt W ., Remarks on superpositionally measurable m ultifunctions (in Russian), Mat. Zamietki ( = Matn. Notes) vol. 48 no 3 (1990), 70 - 72.
O n T H E S U P - M E A S U R A B I L I T Y OF M U L T I F U N C T I O N S . . . 25
Streszczenie
O Supermierzalności multifunkcji o nie koniecznie zwartych wartościach
W tym komunikacie, prezentowanym na konferencji w Czecho-Słowacji, sformułowane zostały kryteria na cięcie multifunkcji 2 zmiennych gwaran tujące jej superpozycyjną mierzalność względem zupełnej sigma-algebry. W odróżnieniu od wcześniejszych prac Caljuka, Spakowskiego i Zygm unta nie wym aga się ciągłości żadnych cięć, lecz zastępuje się ją pewnymi uogól nieniami jednostronnej ciągłości, silną półciąglością z dołu lub warunkiem
( Z ) Grandego. W prezentowanych wynikach nie jest istotne, aby wartości
rozważanych multifunkcji były zwarte. Ponadto przedstawiono 2 kontrprzykłady, wskazujące, że nie jest możliwe dalsze osłabianie założeń w pewnych narzu cających się kierunkach. Pełne dowody znajdują się w [5].
I N S T I T U T E O F M A T H E M A T IC S I N S T I T U T E O F M A T H E M A T C S G D A Ń S K U N IY E R S I T Y P E D A G O G IC A L U N I Y E R S I T Y Gdańsk-Oliwa 80-952 Stwosza 57 Połand Bydgoszcz 85-064 Chodkiewicza 30 Poland
