ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXI (1979) ROCZNIK1 POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO
Séria I: PRACE MATEMATYCZNE XXI (1979)
Alojzy Le c h ic k i (Szczecin)
On continuous and measurable multifunctions
A multifunction F: X -* Y is a function whose value F(x) for x e X is a non-empty subset of Y. Given measure-theoretic or topological structures on X and У, it is possible to define the measurability and continuity of F in various ways.
The purpose of this paper is to investigate of relationships between different measurabilities and continuities of multifunctions, and to obtain in that way certain perspective on the profusion of notions appearing in that context in literature ([1], [3], [4], [7] through [21]). The effort is made to push definitions to their natural limits. Some results are known but considered until now in less generality and dispersed in literature. Therefore, according to the rather expository character of the paper, they are included for the sake of completeness. Also, our intention was to give rather complete proofs — the sometimes tiring style of “that is trivial” being in our opinion incoherent with the general character of the paper.
In Section 1 we recall several definitions for continuity of multifunctions, determine which of them imply the others, and give examples to show which implications do not hold. In Section 2 we consider the question of continuity of the composition q> о F, where (p : Y -> Z is a continuous function and F : X -> Y a continuous, in some sense, multifunction. The results of Section 1 and 2 are used in Section 3 to obtain some generalizations of Lusin type theorems due to Plis [4], Castaing [3] and Jacobs [9].
In Section 4 the problem of the equivalence of the Borel type meas
urability and the Bochner type measurability for multifunctions is considered.
1. Throughout this section all topological spaces are assumed to be Hausdorff and X, Y are topological spaces. We shall denote by stf (У) the collection of non-empty subsets of the space У, by 2y the collection of non-empty closed subsets of the space У, and by %( У) the collection of non-empty compact subsets of У.
If A U . ..,A„ are subsets of У, then ( A t , A „ ) is defined to be the
collection
[ F e 2 Y: F cz (J and At n F Ф 0 , i = 1 , 2 , n}.
i = i
The finite topology on 2y (called also exponential or Vietoris topology) is the topology introduced by means of the base of open collections ( G 1}..., G„>, where Gl , . . . , G n are open subsets of Y ([11], [12]). This topology is the coarsest topology for which the sets ( Z e 2 y: Z c= G} are open whenever G cz Y is open, and the sets { Z e 2 y: Z c= К } are closed whenever K cz Y is closed.
Now, let (У, U) be a uniform space. For F e l l and A cz Y we write V[ A] = {у e У: 3 ( z ,y ) e F } .
zeA
If A = {y} we shall write V(y) instead of F [{y}].
The exponential extension of U to 2y is defined to be the uniformity
2U on 2y determined by the families
{ { A , B ) e 2 Yx 2 Y: F [A ] => В and F [F ] A}, V e i l , as a base of neighbourhoods ([12]).
A multifunction F: X -* 2y is f-upper semicontinuous at x 0 e X ( f-usc at x0e X ) if for each open G containing F( x0) the set ( x e l : F{x) cz G} is open.
F is f-lower semicontinuous at x 0 e X (f-lsc at x0e l ) if for every open G such that F (x 0) n G Ф 0 the set { x e X : F(x) n G Ф 0 } is open. F is f-continuous at x 0 e X if F is /-use and / - lsc at x0e l .
If 2y has the finite topology, then F: X -> 2Y is continuous at x 0 e l iff F is / - continuous at x0e 2 f.
Let (У, U) be a uniform space and let 4/ (x0) denote a base of neighbourhoods of We say that a multifunction F: X - > 2 Y is u-upper semicontinuous at x 0 e X (u-usc at x0e X ) if for every V e U there is C/(x0)g ^ (x0) such that
x e U ( x 0) implies F(x) cz F [F (x 0)].
Dually F: X - + 2 Y is и-lower semicontinuous at x 0 e X (u-lsc at x0e X ) if for every F e l l there is U(x0 )e Ж (x0) such that x e U (x0) implies F (x0) a F [F (x)].
F is u-continuous at x 0 e X if F u-usc and u-lsc at xc0gX.
Let 2y has a topology associated with the uniformity 2U. Then F: X -> 2 y is continuous at x 0e l if and only if F is u-continuous at x 0.
Let (X, 95) be also a uniform space. A multifunction F. X - + 2Y is uniformly u-continuous if for every F e l l there is !Fe3S such that (л^, x2)g IF implies F (x x) c F [F (x 2)] and F (x 2) cz F [F ( x J ].
The grill of ^ ( x 0) ([9]), denoted by # " (x 0), consists of all sets U” (x0) contained in X such that U" (x0) n U (x0) Ф 0 for every U{ xQ)e % (x0).
Continuous and measurable multifunctions 143
For multifunction F: we define p — lim sup F (x) and
p — lim inf F (x) as follows ([9]): x^x°
X ~ > X0
p —lim sup F (x) = П [ U ^(*)]>
U (x0 ) e ^ (x 0 ) x e U ( x 0 )
p — lim inf F (x) = П [ U F(x)].
x ^ x 0 U"( xQ) eJ//” (x o) x e U " (x q)
We say that a multifunction F: X-*. stf{Y) is p-upper semicontinuous at x 0 e X (p-usc at x0 e X ) if p — lim sup F(x) c= F( x0), and that F is p-lower
X-XQ
semicontinuous at x 0 e X (p-lsc at x0e X ) if F( x0) a p — lim inf F (x). A multi- x->x0
function F: X-*-.q/ ( Y ) is called p-continuous at x 0 e X if it is p-usc and p-lsc at x 0.
The above three definitions of continuity for multifunctions were used by Jacobs in [9]. The purpose of this section is to investigate the relation
ship between these notions.
Let (У, U) be a uniform space. We begin by remarking that (of course) each function f : X - * Y may be treated as a “multifunction” F: X ^ 2Y defined by F{x) = { /(x )} , and the following are equivalent
(1) F is u-usc.
(2) F is u-lsc.
(3) F is f-usc.
(4) F is f-lsc.
(5) F is p-lsc.
(6) / is continuous.
But already in that case we have only (6)=>(7) F is p-usc. This is a con
sequence of the following result:
1.1 (Choquet [4]). A multifunction F: X -ks/ ( T ) is p-usc if and only if it has a closed graph G(F) = { ( x j ) e l x f : y € F (x )}.
P roof. Suppose (x0, у0) е X x Y —G(F). Then y0<£F(x0). By assumption there are neighbourhoods V(y0) and U( x0) e aM{x0) such that F(y0) n n U F{x) = 0 - Thus U( x0) x V ( y 0) cz X x Y - G { F ) .
x e U ( x 0 )
Conversely, if G(F) is closed and y0i£F(x0), then there exist neigh
bourhoods F(y0) and U(x0)e^ {x0) such that U(x0) x V ( y0) n G ( F ) = 0 . Hence F(y0) n F ( x ) = 0 for every x e U { x 0), i.e. у0ф р ~ lim sup F(x).
x-x0 1.2. A multifunction F : X -► 2y is f-lsc if and only if it is p-lsc.
P roof. Necessity. Let F b e /-lsc at x0e X . We shall show that F( x0) cz p - lim inf F{x) = П [ U F{x)].
x ^ x 0 U " ( x o)e#"(xo) x e U ” (xQ)
For this, suppose that y e F ( x 0) and let V{y) be an open neighbourhood of y. Then V (y) n f ( x 0) # 0 and by assumption there is [/(x0) e t ( x 0) such that x e U { x 0) implies F (x )n F (y ) Ф 0 .
Noting that U" (x0) n U (x0) Ф 0 and taking x e U " ( x0) n U ( x 0), we obtain x belonging to U” (x0) for which F (x )n V(y) Ф 0 . This means that
y e (J F (x).
x el )" ( xQ )
But U" (x0) n U (x0) Ф 0 for every t/"(x0)e ^ " (x 0); thus
y e П [ U F ( x ) ] -
C/"(x0)6#"(x0) xel/"(*o)
Sufficiency. Suppose that F is not / - lsc at some х0еЛГ. Then there is an open set G such that F (x 0) n G Ф 0 and such that for each U(x0)e'^ (xQ) there exists an x v e U { x 0) for which F i x ^ n G = 0 . Setting U”(x0) = (x^:
U e % ( x 0)} we have U"(x0) n U(x0) Ф 0 whenever U(x0) e Jti(x0). Thus U"(x0)eW"(x0) and F ( x )n G = 0 for each x e U " ( x 0). Let y e F ( x0) r \ G.
G = G (y) being an open neighbourhood of у, уф [J F(x) and F is
not p-lsc. xeU”(x0)
1.3 (cf. [20], Proposition 11). I f a multifunction F: X -► (Y) is p-usc at x Qe X , then the set F( x 0) is closed.
P ro o f. Let y<£F(x0). Then, by assumption, there exist neighbourhoods V(y) and U( xQ) e % ( x 0) such that x e U ( x 0) implies V(y)r\ F(x) = 0 . Hence v(y)n F ( x 0) = 0 , i.e., yi£F (x0). Consequently F (x 0) c- F{ x0).
1.4. Let Y be a regular space. I f a multifunction F: X -> 2* is f-usc, then F is p-usc.
P ro o f. Let y<£F(x0). Then there exist an open set G and open neigh
bourhood V(y) such that F (x 0) c G and V( y) c\ G Ф 0 . F being /-use at XqGA" by definition there is U( x0) e % ( x 0) such that x e U ( x 0) implies F(x) cz G. Thus F ( j ) n F ( x ) = 0 if x e U ( x 0). Therefore
у ф
n [ U f(x)]
[/(xo)eü'(xo) xe U(xQ)
and p — lim su p F (x ) c F{ x 0).
X - + X Q
As an immediate consequence of 1.2 and 1.4 we get the following:
1.5. I f Y is a regular space, then every f-continuous multifunction F:
X -* 2Y is p-continuous.
The converse is not true in general. The following example shows that there exist p-continuous multifunctions which are not /-use.
1.6. Ex a m p l e (cf. [20], Example 4). Let X = R2 and F(x) = F ( x l , x2)
= {(z1,z2)gF 2: z2 = x 2) cz R 2, i.e. the horizontal line through x = (xlv x 2).
Continuous and measurable multifunctions 145
The multifunction F is p-usc. Indeed, let x 0 e X and y<£F(x0). Then p (y ,F (x 0)) = d > 0. Let U(x0) = K ( x0, d / 2), V(y) = K ( y , d / 2). Then we have x e U( x0) which implies dist (F( x0), F (x)) < d/2 and consequently F (x )n n K(y) = 0 . Thus we have the implication уф F( x0)=> уф p — lim sup F(x).
Consequently, p — lim sup F (x) с F (x 0). x_>*°
Now let x0e2 f and y e F ( x 0). For each e > 0 let J^(y) denote the set { z e R 2: q(z, y) < e}. If U"(x0) e ^ " (x 0), then U"(x0) n {x gR2: q(x, x0)
^ e/2} Ф 0 . Hence, looking at the definition of F, we have F{x) ^ K(y) Ф 0 provided x e U " {x0) r\ { x e R 2: q(x,x0) ^ £/2}, i.e., y e p — lim inf F (x).
Thus F is p-lsc. x^x°
Now we are going to show that F is not /-use. Let x0 e R2 be arbitrary and let G = Xq + Kx^ x2) e R 2: \x1x 2\ < 1}. Then the set G is open and F( x0) <= G. But, taking any neighbourhood U of the point x0
and any x e U , x Ф x 0, we have F(x) ф G.
The converse to 1.5 is true under the strong assumption on 7, namely 1.7 (Choquet [4], see also [20], Proposition 8). If Y is a compact space, then each p-usc multifunction F: X-^>2Y is f-usc.
P ro o f. Let G c 7 be an open set for which F (x 0) о G. By assumption for each y e Y —F( x0) there exist open neighbourhoods Vy and Uy (x0)e %( x0) such that x e l / y(x0) implies F { x ) n V y = 0 . The family {G }u {F y: y e 7 —
— F (x 0)} forms an open covering of the compact space 7. Hence there exist П
points y l t ..., yne 7 — F (x 0) such that 7 c G u ( J Vyi. Let U(x0)e$S (x0) be n 1 — 1
a neighbourhood of x0 which is contained in f) Uy.(x0). Then taking x e U (x0)
i = 1
n
we have F (x )n Vy. = 0 for i = 1 , 2 , . . . , и, i.e., F(x) c fl ( Y ~ Vy ) c G.
i — 1
From 1.5 and 1.7 we get immediately
1.8. Th e o r e m. If Y is a compact space, then a multifunction F : X -> 2Y is p-continuous iff F is f-continuous.
1.9. Let (7, U) be a uniform space. I f a multifunction F: X -> 2Y is u-usc, then F is p-usc.
P roof. Since F (x 0) is closed, z ^ F (x 0) implies existence of a F e l l such that z £ F [ F ( x 0)]. Take W e XL symmetric and such that W2 с V. By as
sumption there is G(x0) e ^ ( x 0) such that for x e l / ( x 0) we have F(x) c lF [F (x 0)]. The latter inclusion implies that if y e F ( x ) n W ( z ) , then for some w e F (x 0) (w, y)e W. Also, in that case, (z, y) e W. Hence (z, w) = (z, y)o о ( y , w ) e W2 cz V which is not true. Consequently, F ( x ) n W( z ) = 0 for
x gU(x0) which means also that z фр — lim sup F(x). We have shown that
X - + X Q
^ F (xq) implies z ^ p —lim sup F (x) thus p —lim sup F (x) cz F (x 0).
J C - ^ J C Q X - * ^ X Q
Prace Matematyczne 21.1
As we have remarked at the begining, there exist p-usc multifunctions which are not u-usc (even if F: X-^Y>(Y)).
1.10. Let (У, U) be a uniform space. Every u-lsc multifunction F : X -> 2y is p-lsc.
P ro o f. Let x0e l , U"(x0) e J//"{x0) and K e ll be arbitrary. By assumption there is U{ x0) e # ( x 0) such that x e U ( x 0) implies F(x0) <= F [F (x )]. Since U( x0) n U” (x0) Ф 0 , there exists x e U " ( x Q) such that F( x0) <= F [F (x )].
But it follows that V ( y ) n F ( x ) ^ 0 for some y e F ( x 0). Consequently, y e p — lim inf F(x).
x - > x 0
1.11. Th e o r e m. Let (У, U) be a uniform space. Every u-continuous multi
functions F : X -* 2Y is p-continuous.
1.12. Let (У,11) be a uniform space. Every u-lsc multifunction F : X -> 21
is f-lsc.
This follows immediately from 1.2 and 1.10.
The following example shows that there exist p-lsc (and even p-continuous) multifunctions which are not u-lsc.
1.13. Ex a m p l e. Let X = [0 ,1 ], У = R2 and take the hyperbola ух = 1 therein. Define the multifunction F: X - ^ 2 Y by putting:
For x = 0 F(0) = the non-negative part of y-axis;
For x0 Ф 0 F( x0) = the part of the vertical line through x0 lying between x-axis and the hyperbola.
The multifunction F is p-continuous. We shall prove its p-lower semi
continuity only. Let y = (x0, y )eF (xo), and take a ball K ( y , r) of the point y.
If (/"(x0) e ^ " ( x 0), then x 0 e U” (x0). Hence, clearly, it is possible to find x e U" (x0) so close x0 that the interval F (x) is sufficiently long to intersect К {у, r). Consequently, F is p-lsc.
On the other hand, F is not и-lsc at 0. We have to show that for some s > 0, taking any neighbourhood U of zero, there is x e U such that F (x 0) Ф {z e R 2: q(z, F(x)) < e} = A, where q denotes the metric in R 2. And this is evident since if x Ф 0, A is bounded; hence A cannot contain any semiline.
Let ,^(У) denote the collection of non-empty totaly bounded and closed subsets of the uniform space (У, U).
1.14. If (У, U) is a uniform space and F: X - + 3 î ( Y ) , then the following statements are equivalent:
(a) F is p-lower semicontinuous.
(b) F is f-lower semicontinuous.
(c) F is и-lower semicontinuous.
P ro o f. The validity of (a )o (b ) and (c)=>(b) follows from 1.2 and 1.12, respectively. Implication (b)=>(c) was obtained by Castaing [3J, Theorem 4.1,
Continuons and measurable multifunctions 147
in the special case of metrizable Y and F: X - > (6 (Y). In the case of a uniform space and F: X->.J$(Y) the proof is the same and will be given here for the sake of completeness.
Take V e i l and let W e l l be open in Y x Y and such that W2 <= V.
Since F (xQ) is totally bounded, there exist points x 1, ..., x „ e F (x 0) such that П
F (x0) c= у W(Xi). Hence by assumption there is а (У;(х0)с # ( x 0) (i = 1, 2 ,...
i = 1
..., n) such that х е £ /,( х 0) implies F( x) nW(Xi ) =£ 0 . Let U(x0)e J//(x0) be
ft
a neighbourhood of x0 which is contained in f) и ((х0). Then x e U ( x 0)
i — 1
implies F (x) n W(Xi) ф 0 for i = l , 2 , . . . , n . If y e F ( x 0), then there is 1 ^ i0 ^ n such that y e fF(x,0), i.e., (у, X;0)g W. Since F(x) n lT(xio) ф 0 , there exists a z e F ( x ) for which (xio, z ) e W. Thus (y , z ) = (y, x/o)o(x,-0, z ) e
gW2 cz V, i.e., y e F [ F ( x ) ] . Consequently, F (x0) a V[F(xÿ\ whenever x g g U (x0).
1.15. Let (У, U) be a uniform space. Every f-usc multifunction F: X —>2Y is u-usc.
P roof. Let F e l l be open in Y x Y . Then the set F [F (x 0)] is open.
Thus by assumption there is a neighbourhood U(x0) = { x e X \ F(x)
^ Vl F{ x o)]}.
The converse is not true in general.
1.16. Ex a m p l e. Let F denote the multifunction defined in Example 1.6.
Let us take an arbitrary e > 0 and x0 g! = R2. Then denoting by UE/2(x0) the set {x eR 2: q(x0,x) < e/ 2} we have x g1/£/2(x0)=>F (x) с К1[-Р(*о)]>
where Ve = {(x, y ) ER2 x R2 : g( x , y ) < e}. Indeed, if x gI/£/2(x0) and yeF( x) , then there exists a z e F ( x 0) such that g( y, z ) < e/ 2 . Consequently, F is u-usc at x0 but by Example 1.6 it is not /-use.
1.17 ([3], Theorem 4.1) Let (Y, U) be a uniform space. Every u-usc multifunction F : X ~ > ré ( Y ) is f-usc.
P roof. Let G cz Y be an open set for which F (x 0) <= G. Since the set F (x 0) is compact there is a FgII such that F [F (x 0)] <= G. Indeed, let, for each y e G , Vye U be an open set in У х У such that V2{y) c= G.
The family {Vy (y): y e F { x 0)} covers the compact set F (x 0). Therefore there
n n
is an n e N such that F (x 0) c= (J Еу.(у;) c= G. Taking V = П Vy. e U , we
have F [F (x 0)] c G. i = 1 ' i = 1
By assumption there is a I7(x0) e ^ ( x 0) such that x e U { x 0) implies F(x) c F [F (x 0)] c= G. Thus F is /-use at x 0.
From 1.14, 1.15 and 1.17 we get the following:
1.18. Theorem ([3], Theorem 4.1). If (T,U) is a uniform space, then a multifunction F: X -> ré ( Y) is f-continuous iff F is u-continuous.
As an immediate consequence of 1.8 and 1.18 we get
1.19. Th e o r e m. I f Y is a compact space and F : X -► 2Y, then the following statements are equivalent:
(a) F is p-continuous.
(b) F is f-continuous.
(c) F is u-continuous.
The relationship between the notions of continuity investigated in this section is summarized in the two diagrams below.
/-use
R em ark. Our terminology compares with others as follows:
(a) /-use = strongly upper semi-continuous (Choquet [4]) = weakly con
tinuous (Strother [20]) = upper semi-continuous (Borges [1], Castaing [3], Jacobs [9], Michael [12]) = continuous (Ponomarev [15]).
(b) / - lsc = residually continuous (Strother [20]) = lower semi-continuous (Borges [1], Castaing [3], Choquet [4], Jacobs [9], Michael [12]) = skew- continuous (Ponomarev [15]).
(c) p-usc = upper semi-continuous (Choquet [4]) = cofinally continuous (Strother [20]) = pseudo-upper semicontinuous (Jacobs [9]).
(d) w-usc = upper semicontinuous with respect to inclusion (Jacobs [9]).
(e) м-lsc = lower semicontinuous with respect to inclusion (Jacobs [9]).
2. Let X, Y be topological Hausdorff spaces. We consider the following problem: Given a continuous (in some sense) multifunction F: X- + s t f ( Y) and a continuous function <p: У -* Z , is the multifunction F*: X -> (Z) defined by F* (x) = <p [F (x)] continuous?
2.1. Let Z = <р[У] be the continuous image o f Y by cp. I f a multifunction F: X -* m? (Y) is p-lsc, then the multifunction F* = <poF: 2 f-+ ,c/(Z ) is also p-lsc.
P ro o f. Let x0 e X be arbitrary. Since F*(x0) cz<p{ П
[ U *4*)]}
V "{ xQ )e4 f"( xQ ) x e U ” (x0 )
<= П [
U
[F (x)]] = p —lim inf F* (x),U "(Jco)e^ ' ' ( JC0) x e U " ( x 0) x ^ x o
then F* is p-lsc at x 0.
Continuons and measurable multifunctions 149
A number of examples exist which show that the above proposition is not true for a function F: X -> Y with a closed graph, hence much less for a p-usc multifunction F : X ^ > s / ( Y ) , (i.e. F+ need not be p-usc).
2.2 (cf. [11]). Let Z = ср[У] be the continuous image of Y by cp. If a multifunction F: X -> 2Y is f-continuous, then the multifunction F* = cpoF is also f-continuous provided F^fx) = F^.(x) for every x e X .
P roof. The multifunction F* is / - lsc by Propositions 1.2 and 2.1. Now let us take x 0e X and an open set G => F^(x0) = <p[F(x0)]. By assumption there is a U (x0) e % (x0) such that
x e U (x0) implies F(x) cz q>~1(G).
Hence F^.(x) cz G whenever x e U ( x 0).
The following theorem is given in [12] without proof.
2.3. Let (X, U) and (У, 23) be uniform spaces. A function cp: X -»• Y is uniformly continuous iff the multifunction Ф: I х -* 2Y defined by Ф(А) = ф[А]
is uniformly u-continuous.
P roof. Necessity. Let S { V ) = {{E, F)e 2Y x 2y : V[E] z d F and F [F ] ю E}
be an arbitrary element of the uniformity 2s . Let us take V1 e 23 such that Vf2 œ V. Since the function (p is uniformly continuous there is W e U such that
{xl , x 2) e W implies {(pi xj , <p(x2))e Vl .
Let ( A , B ) e { ( A , B ) e 2 x x 2 x : W[ A ] z d В and IF[B] =з A). If x e A , then there is an element z e B such that { x , z ) e W. Hence (<p(x), <p(z))e Vx and therefore q> (x) e [<p [ £ ] ] . Consequently, (p [A] c [<p [ B ]], but it follows that Ф{А) c= F[<p[B]] = К[Ф(В)]. By a symmetric argument also Ф(В) с Р[Ф(Л)]. Thus (Ф(А), Ф( В) ) её( У) , i.e. the multifunction Ф is uniformly u-continuous.
Sufficiency. Let the multifunction Ф: 2X -> 2Y be uniformly u-continuous.
Then for a given Fe93 there is a W e U such that
W[ A] zd В a 1F[B] 3 A implies У\_Ф(А)] z d Ф(В)а Е[Ф(В)] =э Ф(А).
Let х 1? x 2e X and (xl5 x 2)e W. Then ( { x j , {x2} )e 2 x x2x and {x: } c= IF [{x2}], {x2} c VLQxJ]. Therefore, Ф ^ х ^ ) = {<p(xj)} <= F[{<p(x2)}], i.e., ((pfo), (p{x2))eV.
The next example shows that the continuity (non-uniform) of <p does not in general imply the u-continuity of the multifunction Ф.
2.4. Ex a m p l e. Let the function <p: R -> R be defined by the formula
<p(x) = x 2. Let, for each n ^ 0, an denote the centre of [^/u, yjn + 1 ] . Let
8 = 1/4, A0 = {an: n ^ 0} and <5 > 0 be an arbitrary. Since f n + 1 — yfn -*■ 0 (n -» oo) thus yjn + 1 — yfn < 5 for some n. Denoting by Bô the set
[ y/ n, yjn + 1] u A0, we have
Л0 c V0[_Bf\ = [x: 3 \ x - b \ < <5}
ЬеВА
and
B3 с КДЛ0] = [x: 3 |x —a| < <5).
a e A 0
Since Ф{В0) = Ф(А0) и [n, n + 1] and Ф(Л0) = {af;: n < f l j < w + l , n ^ 0 } , there exists a y e [ u ,u + l ] such that \a% — y\ ^ 1/4. Therefore
Ф(В0) ф {у: 3 |y - z | < 1/4} = 7е [Ф(Л0)].
геФ(Ло)
Consequently, the multifunction Ф is not u-continuous at A0 e 2R.
2.5. Let Z be a uniform space, the image of (У, II) by a uniformly continuous function q>. If a multifunction F: X -> 2y is u-continuous, then the multifunction F: X -> 2Z defined by formula F*(x) = <p[F(x)] is also u-con
tinuous.
P ro o f. Since F*(x) = q>F{x)~\ = Ф оТ(х), then by Proposition 2.3 the multifunction F* is u-continuous as a composition of u-continuous multi
functions.
2.6. Let Z be a uniform space, the continuous image of (Y, U) by (p.
If a multifunction F: X - ^ (6 {Y) is u-continuous, then the multifunction F*:
X- t Yo f Z) defined by the formula F +(x) = <p[F(x)] is also u-continuous.
This follows immediately from 1.18 and 2.2.
3. Throughout this section p denotes a positive Radon measure defined on a e-algebra ,<Z of Borel subsets of a compact metric space X. Suppose that to each finite sequence (h ,...,i„ ) of positive integers there corresponds a subset F,^.. ^ e s # . By E we denote the set
where the union is taken over the collection of all infinite sequences (i„).
If every set E of this form is measurable, we say that the o-algebra .о/
admits the Souslin operation ([18]). Let У be a Hausdorff topological space.
We shall denote by .¥ [6] the collection of closed [open] subsets of У.
A multifunction F: X -> s Z (У) is called -measurable (shortly: measurable) if F -1( T ) = (x e 2 f: F ( x ) n A Ф 0 } e . s / for every A e . F . A multifunction F: X - * . t f ( Y ) is called (l-measurable if F ~ x{ A) e. ^ for every А е й . If У is perfectly normal ([11]), then F-measurability of F implies its C -measurability.
A. P. Robertson has obtained this fact assuming that У is a regular Souslin space and с/ admits the Souslin operation ([18], Theorem 3). If У is a metric
Continuous and measurable multifunctions 151
space and F: X -> Y? (У), then F is measurable if and only if it is (f -meas
urable (Castaing [3]).
We say that a multifunction F : X -> 2Y has Lusin Cp property [ Lusin C f property, Lusin Cu property] if for every e > 0 there is an open set Ee с: X such that p (£ J < e and such that F\X — Er is p-continuous [/-continuous, i/-continuous] ([9]). The notions of Lusin Cp.usc, CHsc, C/usc, Cfhc, Cu.mc, Cu.[x properties are defined in a similar fashion.
The following theorem slightly improves a result of Jacobs [9], Theorem 2.3, obtained for closed valued F into a Polish space.
3.2. Th e o r e m. Let Y be a separable metric space, and F: X ->,о/(У ) a complete valued (i.e. F(x) is complete for every x e X ) multifunction. Then the following statements are equivalent :
(a) F is measurable.
(b) F is (i -measurable.
(c) F has the Lusin Cp property.
P roof. The conclusion follows from [9], Theorem 2.3, and Theorem 2.1 by injecting Y into its completion.
R em ark. A. P. Robertson has obtained the implication (b)=>(a) when Y is a metrizable Souslin space, F\ X -> 2Y, and с/ admits the Souslin operation ([18], Theorem 3).
3.3. (a) Let Y be a topological space. If a multifunction F : X —> 2Y has the Lusin Cf_usc \_Cfllsc or Ср_Ь(] property, then F is measurable [6 -measurable].
(b) Let Y be a uniform space. I f a multifunction F: X -> ^ (T) [£ : X -*■ 2Y]
has the Lusin Cu.usc [Cu_bc] property, then it is measurable [f-measurable].
(c) Let Y be a Polish space. I f a multifunction F: X -* 2Y has the Lusin Cu-usc (or Q-/.sc) property, then F is measurable.
R em ark. Jacobs ([9], Lemma 2.4) proved part (c) of above theorem for F with the Lusin Cu property.
The following theorem generalizes Corollary 2.2 of Jacobs [9]. In his result, F is a separable locally compact space and F has the Lusin CfAx property.
3.4. Let Y be a Polish space. A multifunction F : X - * 2 Y is measurable if and only if it has the Lusin Cf4sc (or Cp_h(.) property.
This result follows from Theorem 3.2 and 3.3(a).
The remaining theorems of this section follow directly from preceding results and Robertson’s Lemma:
3.5 ([18], Lemma 1). Let the a-algebra .о/ admit the Souslin operation.
If Z = <р[У] is a regular Souslin space and F : X ^ 2 Z, then F ~ l (q>\_A]) is measurable for every closed A in Y.
3.6. Let the a-algebra .q/ admit the Souslin operation. I f Z = t p[ Y] is a
regular Souslin space, then every measurable multifunction F: X - * 2 Z has the Lusin Cf_lsc (and Cp4sc) property.
3.7. Let the o-algebra sé admit the Souslin operation and Z = (p[_Y~\
be a metrizable Souslin space. A multifunction F: X -*■ 2Z is measurable if and only if it has the Lusin Cf4sc (or Cp4sc) property.
4. Let X be a set with a er-algebra sé and (У, U) a uniform space.
As in Section 1 we can define a uniformity Ü on s é (У) by taking the families
S( V) = {(A, B) e. s/ (Y)X' B/ ( Y): VIA] з В and V\ B] з A) , V e i l , as a base of neighbourhoods. The symbol ft will also be used to denote the induced uniformity on any subfamily of sé (У). If (У, @) is a metric space, then the topology on sé (Y) associated with Ü is semimetrizable, e.g.
by the Hausdorff semimetric:
h( A, B) = sup { sup q* (у, B); sup q* (y, A) } ,
y e A y e B
where = £?/(l + £). The function h\2Y is a metric.
If А с X , В c s é ( Y) and F: X ^ ^ ( Y ) , then F [A] = {F(x): x e A } and F^(B) = {x e X : F( x ) e B } .
Let rj be a submeasure ([6]) on s é . By J* we denote the set of all multifunctions F: X - ^ .q/ ( Y ) . Let тГе denote the set { A e s / : rj(A) < s}.
Now introduce in ^ the uniform structure which is determined by the families
{ ( F , G ) e . ^ x J F : 3 V (F(x), G (xj) e £> (V)}, V e < & , £ > 0 ,
A e 1 ~ E x e X — A
as a base, of neighbourhoods. The topology on 3F generated by this uniformity is called the topology of convergence in measure and is denoted by rn. If ( У, @) is a metric space, then the topology is semimetrizable by the écart ([2]) dn defined as follows:
dn(F, G) = inf {rj(A)+ sup h [ F (x), G (x)]}.
x e X - A
If rj* is the Jordan extension of rj, i.e., rj*(E) = inf [rj(A)\ E - c z A e s t } , then (see e.g. [5])
d„*( F, G) = inf {rj*(A)+ sup h [F(x), G(x)]} = d (F, G).
A Œ X x e X - А
The equivalent écart on & may be given, for instance, by the: formula (cf. [13])
4}(F ,G ) = inf {s + rj *( {xeX: h [F (x), G(x)] > e})}.
£ > 0
A multifunction F e # - will be called s é -simple (or simple) if it assumes
Continuous and measurable multifunctions 153
only a finite number of values A x, A 2, ..., Ake, z/ ( Y) , each on a measurable set, that is, on the set X t = { x e l : F( x) = /4f} e j / for i = 1 , 2 , . . . , A: and
U X i = X . i = 1
Multifunctions belonging to the closure (in the topology t,,) of the set of all .c/-simple multifunctions in fF are called ц-measurable (or Bochner measurable).
Let (Y, g) be a metric space. A sequence {Fn)neS c= is called pointwise convergent if for every x e X the sequence (F„ (x))„eiV is convergent in the space (j/ (Y) , h). A sequence (F„)neN cz FF is said to be ц-uniformly convergent to F e /F if for every s > 0 there is a set £ e , j / such that F„(x)->F(x) uniformly on X - E and rj(E) < e.
Many classical theorems are valid for sequences of multifunctions. In particular (cf. [5]):
4.1. Let g be a o-submeasure on ,</. I f Fn- * F (n-+ oo) in measure (i.e. in the topology of convergence in measure), then there is a subsequence of (Fn)neN which is ц-uniformly convergent to F.
4.2. If ц is a o-submeasure on .</ and Y is complete, then the space of all ц-measurable multifunctions is complete with respect to convergence in measure.
4.3 (Egoroff). Let ц be an order continuous submeasure ([6]) on ,</.
A sequence (Fn)nsN of ц-measurable multifunctions is ц-uniformly convergent to F if and only if Fn (x) -> F (x) ц-almost everywhere on X.
4.4. Co r o l l a r y. Let ц be an order continuous submeasure on .я/. A multi
function F is ц-measurable iff there is a sequence (Fn)neN o f s é -simple multi- functions such that Fn(x) -* F (x) ц-almost everywhere on X.
C. Castaing and M. Q. Jacobs investigated relations between Bochner measurability and J^-measurability and have proved the following theorems:
4.5 (Castaing [3], Theorem 4.9). Let X be a locally compact space, p a Radon measure on X , and Y a metrizable compact space. If a multi
function F: X - * (F( Y) is .^-measurable, then there is a sequence (Fn)neN of simple multifunctions which is uniformly convergent to F.
4.6 (Jacobs [9], Corollary 2.5). Let X be a compact metric space, p a Radon measure on X , and (Y, g) a metric separable locally compact space. Let the space 2Y be equipped with the topology generated by the metric of the one-point compactification of Y. A multifunction F : X ^ 2 Y is .9r-measurable if and only if there is a sequence (F„)neN of simple multi
functions converging to F p-almost everywhere.
The proofs of the above theorems were based on the Lusin theorem.
4.7. Let (F„)neN be a sequence of .^-measurable multifunctions Fn: X -* Y>{Y) pointwise converging to F: X —► Y>'{Y). Then the multifunction F is iF-meas
urable.
P roof. Let А с У be a closed set. We claim that F *(/4) = [x gX:
V 3 V Р „ [Ы х )]п Л ф 0 ] , where V„ = { ( y , z ) e Y x Y : g( y, z ) < l /и].
>!■ V m e / V m
Let x e F -1(A) and n e IV be arbitrary. By assumption there is an m e IV such that F(x) cz K,[Ffe(x)] for к ^ m. Thus K,[Fk( x )] n T Ф 0 for к > m.
Conversly, let j s N be arbitrary. Take an nelV such that Vn2 c I/•. There is an m e IV for which Fk(x) cz Vn [F(x)] and V„ [F k(x)] n A / 0 provided к ^ m. Then K,[Fk(x)] <= К [_K DF(*)]] cz T/[F(x)] and it follows that
n F (x) Ф 0 . Since the set F (x) is compact and F (x) n Vj [Л] zz F (x) n
00
n Vi[A] for i > j , F ( x ) n A — П [ F (x ) n Р/[Л ]] Ф 0 - Consequently F _1(A)
*■ j'= i
00 00 00
= n U П f f H K I A ^ g.ç/ .
n = 1 m=1 fc= m
Let .c/ 0 be the Lebesgue extension of .с/ and let >/0 denote the Lebesgue extension of rj on ,o/0. We recall that the spaces of r\- and
^о-measurable multifunctions are identical. Moreover, both the convergences in measure (r\ and r\0) are equivalent.
The proof of the following theorem is inspired by, and similar to, a proof of Drewnowski [5], Theorem 3.5.
4.8. Theorem. Let ц an order continuous submeasure on .с/ and (Y, a metric space. A multifunction F : X - + f6 (Y) is Bochner measurable if and only if it is (о/ 09 zFfmeasurable and almost separably valued, i.e., the set F^X — £ ] is separable for some r]0-null set E.
P roof. Necessity. By Theorem 4.1 there is a sequence (Fn) of simple multifunctions which is ^-uniformly convergent to F. Thus there is a set Ее.я/ such that rj(E) — 0 and F „ (x )-> F (x ) for x gX — E, i.e. ^-almost everywhere. Since FI X — Е] cz (J Fn [X— E] (where the closure is taken in
n= 1
(2Y,h)) and the sets F„[W —£ ] (n e N) are finite, the set F [ X — E] is sepa
rable. If A is a closed subset of Y, then by 4.7 we have F ~1(A)
= ( F -1( A ) n E ) u ( F ~1( A ) - E ) G ^ o -
Sufficiency. Let E be a rj0-null set such that the set F [ X —E] is separable.
Let us take a fixed element B0g%(Y) and an arbitrary s > 0. There is a sequence (Bn)neN which is dense in { AgF [ X — £ ]: h(B0, A) > ej. Define the sets A„( nGN) as follows: An = [x gX — E: h( F( x) , B„) < e and h(F{x), B0) > ej. By [3], Theorem 4.3, we have Л„е.с/ 0 for n e N. Assume C x = A x and C„ = An — ( Al u ... и An-1) for n ^ 2. Let nKeIV be such that
00
»/o( U Q < v. Define a multifunction F, : X ->7I(T) by the formula n = nr + 1
( Bn for xeC „, n ^ n,,, ) B0 for x e l - 0 Cn.
к n= 1
F r. (a) =
Continuous and measurable multifunction 155
The multifunction Fe is .o/0-simple. Moreover, ( x e Z : h ( F(x), FE(x)) > i:j
QO 00
c E u (J C„. Indeed, if x$ Ekj IJ C„, then h(F(x), FE(x)) = h(F(x), Bn)
n = n f: + 1 n —nr + 1
00
< f for some 1 ^ n ^ nE, or h(F(x), B0) ^ г and хф IJ C„, i.e. h(F(x), n — 1
Fr(x)) — h(F(x), B0) ^ г. Consequently, rç*({xeX : h(F(x), FE(x)) > e]) < e, i.e. F is Bochner measurable.
4.9. Co r o l l a r y. Let rj be an order continuous submeasure on xY and let Y be a separable metric space. A multifunction F: X - ^ (6 (Y) is Bochner measurable if and only if it is (.c/0, .^-measurable (equivalently : (.?/0, C ^me
asurable).
This follows immediately from 4.8 and [3], Lemma 4.1.
The author is very much indebted to Doctor Iwo Labuda for helpful remarks and conversations.
References
[1] C. J. R. B orges, A study of multivalued functions. Pacific J. Math. 23 (1967), p. 451-461.
[2] N. B o u rb a k i, Topologie générale, Ch. IX, Paris 1958.
[3] C. C a sta in g , Sur les multi-applications mesurables, Rev. Française Informatique Recherche Opérationnele 1 (1967), p. 91-126.
[4] G. C h o q u e t, Convergences, Grenoble Univ. Annal. 23 (1947), p. 57-112.
[5] L. D re w n o w sk i, Certain problems from the theory of set functions and integrable functions [in Polish], Thesis, Poznan 1970.
[6] —, Topological rings of sets, continuous set functions, integration, I, II, III, Bull. Acad.
Polon. Sci., Sér. Math., Astr., Phys. 20 (1972), p. 269-276, 277-286, 439-445.
[7] S. E ile n b e rg and D. M o n tg o m e ry , Fixed point theorems for multivalued transformations, Amer. J. Math. 68 (1946), p. 244-222.
[8] C. J. H im m e lb e rg and F. S. Van Vleck, Some selection theorems for measurable functions, Canadian J. Math. 21 (1969), p. 394-399.
[9] M. Q. Ja c o b s, Measurable multivalued mappings and Lusiri’s theorem, Trans. Amer. Math.
Soc. 143 (1968), p. 471-481.
[10] S. K a k u ta n i, A generalization of Brouwer’s fixed point theorem, Duke Math. J. 8 (1941), p. 457-459.
[11] K. K u ra to w s k i, Topology I, Academic Press-PWN, New York 1966.
[12] E. M ic h ae l, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), p. 152-182.
[13] C. O lech , A note concerning set-valued measurable functions, Bull. Acad. Polon. Sci., Sér. Sci. Math., Astr., Phys. 13 (1965), p. 317-321.
[14] A. P lis, Remark on measurable set-valued functions, ibidem 9 (1961), p. 857-859.
[15] V. I. P o n o m a re v , A new space o f closed sets and multi-valued continuous mappings of bicompact a, Amer. Math. Soc. Transi. 38 (1964), p. 93-118.
[16] W. Pu, Another Ascoli theorem for multi-valued functions, Comm. Math. 17 (1974),
p . 445-450.
[17] L. R a tn e r, Multi-valued transformations, University of California, 1949.
[18] A. P. R o b e rts o n , O.n measurable selections, Roy. Soc. Edinburgh Proc. A72 (1974), p. 1-7.
[19] R. E. S m ith so n , Some general properties of multivalued functions, Pacific J. Math. 15 (1965), p. 681-703.
[20] W. L. S tro th e r , Continuous multi-valued functions, Boletim Soc. Math. S3o Paulo 10 (1955), p. 87-120.
[21] A. D. W allace, Cyclic invariance multi-valued maps, Bull. Amer. Math. Soc. 55 (1949), p. 820-824.
ZAKEAD MATEMATYKI, WY2SZA SZKOLA PEDAGOGICZNA, SZCZECIN
