On approximation by step multifunctions

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXV (1985) Series I: COMMENTATIONES MATHEMATICAE XXV (1985)

An d r z e j Sp a k o w s k i (Opole)

On approximation by step multifunctions

Abstract. We consider a pointwise approximation of mutlifunctions by nets of semicontinuous step multifunctions. Moreover, two lemmas about intersection of lower semicontinuous multifunctions are proved.

1. Introduction. Let X and У be non-empty sets and denote by a (Y ) the family of all subset of Y and by a 0 (Y ) the family of all non-empty subsets of У Every mapping F : X -> а (У) is said to be a multifunction. A multifunction will be called a step multifunction if the set of its values is finite.

Beer [1] has considered approximation of semicontinuous multifunc­

tions, defined on a rectangular parallelepiped P c= R" with values in a metric space, by sequences of semicontinuous step multifunctions. In this paper we examine this problem for multifunctions defined on an arbitrary topological space. It will shown that in this case semicontinuous multifunctions can be approximated by nets of semicontinuous step multifunctions.

Now, we recall some definitions and facts concerning continuous multifunctions (see [8], [7]).

Let X and У be topological spaces. A multifunction F from X to Y, i.e., F : X -> а (У), is /-upper semicontinuous at x 0 e l if for each open G a Y containing F { x 0) the set { x e X : F (x ) c: G} is open in X. F is /-lower semicontinuous at x 0 e X if for each open G c Y such that F (x0) n G Ф 0 the set ( x e l : F ( x ) n G Ф 0 } is open in X. F is /-upper semicontinuous if it is /-upper semicontinuous at every x e X . F is /-lower semicontinuous if it

is /-lower semicontinuous at every' x e X .

Assuming that У is a quasi-uniform space and ЩУ) is its quasi­

uniformity we obtain two notions of u-semicontinuity. Note also that each topological space is quasi-uniformizable (see [9]).

A multifunction F from X to У is м-upper semicontinuous at x 0e X if for each W eU (Y ) there is a neighbourhood U of x 0 such that F (x ) a W ( F(x0)) for every x e U , where IL (F (x 0)) denotes the set {y e Y:

(z, y ) e iy f o r some z e F ( x 0)}. F is м-lower semicontinuous at x 0 e l if for each We U(Y) there is a neighbourhood U of x 0 such that F (x 0) c= W ~ 1(F(x)) for every x e U, where W ~ 1 — {(x, y)e Ух У: (y, x)e W }.

F is w-upper semicontinuous if it is w-upper semicontinuous at every ^{x e X .} F is w-lower semicontinuous if it is w-lower semicontinuous at every ^{x e X .}

The following facts hold: /-upper semicontinuity implies w-upper semicontinuity, м-lower semicontinuity implies /-lower semicontinuity,/-lower semicontinuity implies w-lower semicontinuity of a multifunction F : X -►а0(У) provided У is a uniform space and all the F (x ) are closed and totally bounded.

Let {F t, t e T } be a net of multifunctions from X to Y, where У is a quasi-uniform space. The net is pointwise convergent to a multifunction F : X -*■ a(Y ) if for every x e X and every We Щ У) there exists t0 E T such that, for every t ^ t0, F t (x) cz W (F(x)) and F (x ) =э W ~ x (Ft(x)). Then we will write: F t(x )-> F (x ) for every ^{x e X .}

A multifunction F from X to Y, where X and У are topological spaces, is subcontinuous (see [10]) if whenever {x(, t e T ) is a convergent net in X and {y,, ^{t E} T) is a net in У such that yte F ( x t) for all ^{t E} T, then the net (yt, ^{t E} T} has a convergent subnet. Note that if У is a compact space, then each multifunction from X to У is subcontinuous.

Finally, the graph of a multifunction F from X to У is the set G (F )

= { ( х , у ) е * х У : y e F ( x ) } .

2. General remarks. Let X be a set and У be a quasi-uniform space.

A net {F t, t e T} of multifunctions F t: X -» a(Y ) will be called an upper approxim ation (resp. a lower approxim ation) for a multifunction F : X -> а (У) if it is pointwise convergent to F and for every x eX and tx ^ t2 the inclusions F (x ) c= F t2(x) <= F t (x) (resp. F, (x) <= F t2(x) c= F (x)) hold.

2.1. Pr o p o s i t i o n. Let X be a topological l\-space and Y be a quasi­

uniform space. Then fo r arbitrary multifunction F from X to Y

(a) there exists an upper approxim ation f o r F by f-low er semicontinuous step multifunctions,

(b) there exists a lower approxim ation fo r F by f-upper semicontinuous step multifunctions.

P r o o f, (a) Let T b e the family of all finite subsets of X upward directed by the inclusion. We define F ((x) = T (x) if ^{X E t} and T t(x) = У if x ^ f, for all

î e T and ^{x e X .} The net {F t, t e T ) is the required upper approximation.

(b) Let T b e as above and define F t(x) = F (x ) if X E t and F t(x) = 0 if x $ t , for all t E T and x eX. The net {F t, t e T } is the required lower approximation.

Note that the approximation defined in the above proof have the following property: for each ^{x e X} there is ^{t 0 e T} such that F t(x) — F (x ) for t ^ t 0, i.e., T f (x )-> T (x ) in the discrete uniformity on У

R e m a rk s , (a) There exists a multifunction with no upper approximation by w-upper semicontinuous multifunctions. Indeed, let X — Y

= [0, 1] and define F (x ) = 0 if x is irrational and F (x ) = [0, 1] if x is rational. If {F t, t ç T ) is any upper approximation for F, then for each irrational x oe [ 0 , 1] there exists t0 with F tQ(x0) = 0 and of course F tQ(x)

= [0, 1] for all rationals x. But such multifunction is not u-upper semicontinuous at x 0.

(b) (due to Beer [1]). There exists a multifunction with no lower approximation by /-lower semicontinuous step multifunctions. We take X = Y = [0, 1] and define F {x ) = {x} for all x e [ 0 , 1].

2.2. Pr o p o s i t i o n. L et X be a topological space, Y a quasi-uniform space and let F be a multifunction from X to Y.

(a) I f F is compact-valued, then there exists a lower approxim ation fo r F by subcontinuous step multifunctions.

(b) I f X is a com pact space, Y a regular space and F is subcontinuous, then there exists an upper approxim ation fo r F by subcontinuous step multifunctions.

P r o o f, (a) The proof is the same as the proof of Proposition 2.1(b).

(b) By the hypotheses the set F (X ) = (J F(z) is relatively compact ([3],

zeX

Corollary 5.7). Now, let T be as in the proof of Proposition 2.1. For t e Г let F t(x) = F ( x) if x e t and F t{x) = F (X ) if x $ t . The net {F t, t e T } is the required upper approximation.

R e m a rk s , (a) The compactness of the values F (x ) in Proposition 2.2(a) can not be omitted. For example, if we take the multifunction F (x ) = [0, 1]

for all x e [ 0 , 1] except x = 1/2 and F(l/2) = [0, oo), then F has no lower approximation by subcontinuous multifunctions.

(b) The compactness of the space X in Proposition 2.2(b) is essential.

Indeed, if we take X = (0, 1), У = [ 1 , oo) and define F (x ) — [1, 1/x] for x e (0 , 1), then F is subcontinuous but if F t is any step multifunction on X with Т Д х) => F (x), then there exists x oe (0 , 1) such that F i (x) = [1, oo) for all x e (0 , x 0) and the multifunction F t is not subcontinuous.

In the next two sections we use a slightly generalized Beer’s method of approximation to obtain some results concerning approximation by nets. In the metric case our results generalize those of [1]. First, we will generalize the construction of the so-called blocks (see [1]).

Let X be a topological space and A = {D x, . .. , Dn} be a partition of X, i.e., I = l ) 1 u ... u D „ and the sets Д are pairwise disjoint. The sets В, = Д , i = 1, . . . , n, where Д is the closure of Д in X, we will call blocks of the partition A. For each x eX let B (x) be the family of all blocks В of A such that x e B . Now, we define an equivalence relation on X as follows: x ~ у if and only if B (x) = B(y), i.e., if x and у belong to the same blocks of the partition.

12 — Prace Matematyczne 25.2

In Sections 3 and 4 we will consider an approximation by step multifunction which are constant on equivalence classes of the relations defined as above for given partitions.

3. Upper approximation. We prove two theorems on the approximation of w-upper semicontinuous multifunctions by nets of /-upper semicontinuous step multifunctions with closed values. We will also remark that in the metric case the approximation by sequences is possible.

Recall that a locally compact space X is cr-compact if it is a countable union of compact subspaces X n, n e N such that X„ a int X n+l for each n e N (see [5], p. 250). We may also assume that int X 1 # 0 .

3.1. Th e o r e m. L et X be a locally com pact and o-com pact space, Y be a uniform space and F : X -» a 0 (Y) be a u-upper semicontinuous multifunction.

Then F has an upper approxim ation by f-upper semicontinuous closed-valued step multifunctions.

P ro o f. By the hypotheses there exists a sequence { X „ ,n e N } of 00

compact subspaces of X such that X = (J X n, int X 1 Ф 0 and

n= 1

X„ c=intA/ + 1 for each n e N . Let U (X ) be any quasi-uniformity of X. For each n e N and Fell(A ") let P(n, V) be the family of all partitions A of X n such that for each block В of A there exists y e X such that В cz V{y). Let T = {(n, V, A): n e N , F e l l (A) and A e P ( n , V)}. T is a directed set with the following relation: (n, V, A) ^ (n', V , A') if and only if n ^ n\ V cz V and A' restricted to the subspace X n is a subpartition of A. Now, for n eN , FeU(Af) and each block В of a partition A e P { n , V) we define the following multifunction on X:

9B{x) = F ( B) if x e B and 0e (x) = 0 otherwise. Then the net {F t, t e T }, where

F t(x) = (J 0B(x) if x e in tX „ and F t(x) — Y в

otherwise, t = (n, V, A) and the summation is taken over all blocks В of A, is the required upper approximation. Indeed, it is obvious that each F t is a closed-valued step multifunction and for each x e X and t ^ t' we have F (x) cz F t.(x) cz F t(x). Multifunctions F t are /-upper semicontinuous since multifunctions 0B are such and the sum (J is taken over a finite set.

в

It remains to prove the convergence. Let x 0 e X and IF e ll(T ). We take n0e N and V0 e U(Ar) such that x 0 e in tX „ 0 and for each block В э х 0 of each partition A e P { n 0, F0) there holds F (B) cz U (F (x 0)), where l/ eU (T ) and U о U c W. Therefore we have

e B(x0) a U o U ( F ( x 0)) cz W(F(xo))

and hence F ,0 (x0) cr ^ ( F ( x 0)), where t0 = ( n 0, V0, A). This completes the proof.

3.2. Th e o r e m. L et X be locally com pact and o-com pact space, Y be a quasi-uniform regular space and F : X a 0 {Y) be a u-upper semicontinuous closed-valued multifunction. Then F has an upper approxim ation by f-upper semicontinuous closed-valued step multifunctions.

P r o o f. We start as in the proof of Theorem 3.1 and for each block В we define

0B(x) — F ( B) if x e B and вв (х) = 0

otherwise. Then the net {F t, t e T } defined as in the proof of Theorem 3.1 is the required upper approximation. Indeed, the multifunction F restricted to the set В has closed graph (see [6], p. 175). Now, by a result of Choquet ([2], p. 66) the set F (B ) is closed.

It remains to prove the convergence. Let x 0 e X and кРеЩУ). We take n0 e N and T 0êU ( I ) such that x 0 Gint2f„0 and F (В) a W (F (x0)) for each block В of each partition A e P ( n 0,V 0). Then F t0(x0) c= IT (F (x 0)), where

= (ио> V0, A). This completes the proof.

R e m a rk s , (a) If in Theorems 3.1 and 3.2 we assume also that the space X is metrizable we may prove that there exists the approximation by sequences of/-upper semicontinuous closed-valued step multifunctions in the following way. Let A 1 be any partition of the subspace X 1 with blocks of diameter less than 2 ~ 1. For each n > 1 let A„ be a partition of X„ with blocks of diameter less than 2~" such that A„ restricted to the subspace X n. 1 is a subpartition of the partition A „ -1. Finally, we define the following sequence of step multifunctions on X :

F„(x) = (J 0g (x) if x e in tX „ and F„(x) = У в

otherwise.

(b) If in Theorems 3.1 and 3.2 the multifunction F is compact-valued, then there exists an upper approximation by compact-valued step multi­

functions since the image of a compact set by a compact-valued /-upper semicontinuous multifunction is compact (see [10]) and for compact-valued multifunctions /-upper semicontinuity is equivalent to the м-upper semiconti­

nuity (see [8]).

4. Lower approximation. In this section we will show that a lower approximation by /-lower semicontinuous step multifunctions is possible under additional assumptions (see [1], p. 14). Remark also that the Beer’s proof is not complete since /-lower semicontinuous multifunctions are not, in general, closed under finite intersection (see [5], p. 180). However, his result is true since we have the following two lemmas.

4.1. Le m m a. Let X be a topological space, Y be a normed space and assume that a multifunction F from X to Y is и-lower semicontinuous at x 0 e X . I f int F (x0) Ф 0 and F has closed and convex values on a neighbourhood o f x 0, then there exists a neighbourhood U o f x 0 such that int f) F (y) # 0 .

y e U

P ro o f. Let s e in t( F ( x 0)). There exists a bounded neighbourhood К of 0 in У such that s + V + V a F ( x 0). Because F is u-lower semicontinuous at x 0 then there exists a neighbourhood U of x 0 such that for every y e U the inclusion F ( x 0) <= F (y )+ V holds. Therefore for every y e U we have

s + V + V c z F (y )+ V .

We may assume that for every y e U the set F(y) is closed and convex.

By the law of cancellation (see [4]) we obtain that for every y e U the inclusion s + V a F (y) holds. In other words s + V a f) F (y) which ends

the proof. yeU

4.2. Le m m a. L et X be a topological space and Y be a finite dimensional space. Assume that multifunctions F x and F 2 frtpn X to Y are u-lower semicontinuous at x 0 e X and have closed and convex values on a neighbourhood o f x 0. Then the intersection multifunction F = F x n F 2 is и-lower semicontinuous at x 0, provided F ( x 0) is com pact and int F ( x o) ^ 0 .

P ro o f. Let F be an arbitrary neighbourhood of 0 in У and Z (V ) be a finite subset of in tF (x 0) such that for every ^{z e F (x}0) there exists s e Z ( V ) with ^{z — s e V .} For every ^{s e Z ( V )} let Vs be a neighbourhood of 0 in У such that s+ V s c z F (x0). The set

v0= П К

seZ(V)

is a neighbourhood of 0 in У such that for each / = 1 ,2 we have Z (V )+ V0 c= F (x0) c: F j(x 0). Take an arbitrary ^{ze F (x0 ) .} There exists s e Z ( V ) with ^z — ^{s e V} Since F x and F 2 are u-lower semicontinuous at x 0 then there exists a neighbourhood U of x 0 such that F t (x0) c Fi (y) + Lo f° r every ye U and / = 1 ,2 . Therefore for every y e l/ and / = 1 ,2 the following inclusion s + F0 c (y) + K> holds. We may assume that V0 is bounded and the sets F fy ) , y E U , / = 1 ,2 are closed and convex. By the law of cancellation we get se F ,(y ). Hence for all y e U we have se F (y ). This implies that for all y e U, z ^{= s + z — s e} F (y)+V, which ends the proof.

Now, we are ready to prove the following approximation theorem.

4 3 . Th e o r e m. L et X be a com pact H au sdorff space, Y be a finite dimensional space and F : X -> a 0 (Y) be an f-low er semicontinuous multifunction. I f the values o f F are com pact convex bodies, then F has a lower approxim ation by f-low er semicontinuous step multifunctions with values which are com pact convex bodies.

P ro o f. Let ll(2Q be the uniformity of X. For each VeVL(X) we define the set P(V ) to be the family of all partitions of X with blocks В satisfying В cz V(y) for some y e X . Let T be the set {(V, A): F e ll( X ) and A e P (V )}. Tis a directed set with the following relation: (V, A) ^ (V', A') if and only if V cz V and A' is a subpartition of A. For each block В we define the following multifunction on X :

9B^{(x) =} 0 F(y) if x e В and 9B(x) = Y

yeB

otherwise. Next, for each t = (V, A ) e T and x e X we define F t(x) = f ] 9 B(x),

в

where the intersection is taken over all blocks В of A. We will show that a subnet of {F t, t e T } is the required lower approximation. It is obvious that the multifunctions F t are step multifunctions and F f l (x) c= F t (x) <= F (x ) for every x e X and tl ^ t 2. Moreover, the multifunctions 9B are /-lower semicontinuous.

We claim that there exists t0 e T such that in tF t(x) Ф 0 for every t ^ t0 and x e X . First, let us observe that there are FgU (Z ) and a finite subsët {x 1? . .. , x„} cz X such that X = K ({x l5 . . . , x„}) and

int П F ( y ) # 0 у6к(й:дсг))

for every i = 1, . . . , n. Otherwise we could construct a net {x„, V eU {X )} such that

( - ) int П F {y) = 0 .

yeV (V {xv ))

Since X is compact, the net {x v , F e U (X )} has a cluster point x + e X . By Lemma 4.1,

int

П

for some V+ е Щ Х ). But taking F + e U ( I ) with V + o V + c V+, we can find WgU(X) such that W o W a V + and х ж е К + (х +). Hence W o W (x w) cz K+(x+) and consequently

0 = int П F ( y ) c=int П F (y),

yeV + ( x + ) yeW (W (xw ))

which contradicts ( —). Now let us take a partition A e P {V ) such that every block В of A is included in a set F(x,), i e {1, n}, and В cz F(y) for each y e B . Writing t0 = (V , Â) we can easily verify that in tF t(x) Ф 0 for every t ^ t0 and x e X.

Moreover, all the sets F t (x) are convex and compact. Now, by Lemma 4.2 we obtain the n-lower semicontinuity of the multifunctions F t.

It remains to show the convergence. Let x 0 e X and W be an arbitrary neighbourhood of 0 in Y By the w-lower semicontinuity of F there exists Vx c ll(X ) such that, for every z e V i (x0), F ( x 0) <= F {z) + W. We may suppose, applying Lemma 4.1, that

int П F (z) = £ 0 .

zeV ^{± {x q)}

We will show that there exists V0 eVL(X) with V0 c V1 such that

(1) F ( x 0)cz П F (z )+ W .

zeK0(x0)

Let Z be a finite subset of int F ( x 0) such that F ( x 0) cz Z + W . Then there exists a bounded neighbourhood V of 0 in Y satisfying Z + V cz F (x0). By the м-lower semicontinuity of F there exists F0eU ( I ) with V0 cz V1 such that F ( x 0) c z F (z )+ V for every zeV 0 (x0). Thus for every z e V 0 (x0)

Z + K c F ( x 0) c F ( z ) + F and by the law of cancellation we get

Z c П F(z)

z eV 0 (x 0 )

and

F (x0) cz Z + W а П F (z )+ W ,

zeV q(x q)

i.e., condition (1) holds.

Finally, if we take t0 = (V0, A0), where A0 e P (V 0), then F ( x 0) cz F tQ (x0) + W. This ends the proof.

R e m a rk . If in Theorem 4.3 we assume that the space X is metrizable, then we can construct a lower approximation by sequence of step multifunc­

tions using partitions with blocks of diameter less than 2~", n e N (see Section 3).

I am much indebted to Dr. A. Lechicki for his valuable remarks concerning the first version of this paper.

References

[1] G. B eer, The approximation o f upper semicontinuous multifunctions by step multifunctions, Pacific J. Math. 87 (1980), 11-19.

[2] G. C h o q u et, Convergences, Grenoble Univ. Annal. 23 (1947), 57-112.

[3] S. D o le ck i, G. H. G re co , A. L e c h ic k i, Compactoid and compact filters, to appear.

[4] L. D rew n o w sk i, Additive and countably additive correspondences, Comment. Math. 19 (1976), 25-54.

[5] R. E n g e lk in g , General topology, PWN, Warszawa 1977.

[6] K. K u ra to w s k i, Topology, I, Academic Press-PWN, New York 1966.

[7] A. L e c h ic k i, On continuous and measurable multifunctions, Comment. Math. 21 (1979), 141-156.

[8] —, Some problems in the theory o f continuous and measurable multifunctions (in Polish), Thesis, University of Poznan, 1980.

[9] W. J. P erv in , Quasi-uniformization o f topological spaces, Math. Ann. 147 (1962), 316-317.

[10] R. E. S m ith so n , Compactness and connectedness as composable properties, J. Austral.

Math. Soc. 18 (1974), 161-169.

[11] —, Subcontinuity o f multifunctions, Pacific J. Math. 61 (1975), 283-288.

INSTYTUT MATEMATYKI, WY2SZA SZKOLA PEDAGOGICZNA OPOLE, POLAND