DECOMPOSABLE HULLS OF MULTIFUNCTIONS Andrzej Nowak

DECOMPOSABLE HULLS OF MULTIFUNCTIONS Andrzej Nowak

Silesian University Institute of Mathematics Bankowa 14, 40–007 Katowice, Poland

e-mail: anowak@ux2.math.us.edu.pl and

Celina Rom University of Bielsko–BiaÃla Department of Mathematics Willowa 2, 43–309 Bielsko–BiaÃla, Poland

Abstract

Let F be a multifunction with values in L

(Ω, X). In this note, we study which regularity properties of F are preserved when we consider the decomposable hull of F .

Keywords: decomposable set, multifunction, decomposable hull.

2000 Mathematics Subject Classification: 26E25, 49J53, 54C60.

1. Definitions and some elementary properties

In this section, we give the notation and definitions used in the whole paper.

Let (Ω, Σ, m) be a measure space and (X, k·k) a separable Banach space. By L

(Ω, X) we mean the Banach space of equivalence classes of Σ-measurable functions v : Ω −→ X with the norm kvk

= ( R

Ω

kvk

dm)

< +∞ for

1 ≤ p < +∞, and kvk

= ess sup

kv(ω)k < +∞ for p = +∞. An open

ball in L

(Ω, X) with the center v and radius r is denoted by B(v, r). For

A ∈ Σ we denote by χ

the characteristic function of A. We call a set K ⊂

L

(Ω, X) decomposable if for all u, v ∈ K and each A ∈ Σ, χ

u+χ

v ∈ K

(cf. Hiai and Umegaki [1], Olech [3]). Let A ⊂ L

(Ω, X). By a decomposable

hull of A, dec A, we mean the smallest decomposable subset of L

(Ω, X) which contains A, i.e.

dec A := \

{K : A ⊂ K and K ⊂ L

(Ω, X) decomposable}.

A closed decomposable hull of A, cl dec A, is the smallest closed and decom- posable subset of L

(Ω, X) containing A. It is not difficult to see that

dec A = n X

i=1

χ

v

: A

∈ Σ, v

∈ A, i = 1, . . . , n, [

A

= Ω, A

∩ A

= ∅

for i 6= j, n ∈ N o

and cl dec A = cl (dec A), where cl denotes the closure operation. Such a representation of elements from dec A will be used in the whole paper.

Let T, Y be nonempty sets. By P(Y ) we denote the family of all nonempty subsets of Y . A mapping F : T −→ P(Y ) is called a multi- function from T to Y . For Z ⊂ Y we define

F

(Z) = {t ∈ T : F (t) ∩ Z 6= ∅}, F

(Z) = {t ∈ T : F (t) ⊂ Z}.

In this paper, we study multifunctions from a topological or measurable space T into L

(Ω, X). We associate with F : T −→ P(L

(Ω, X)) two new multifunctions G(t) := dec F (t) and G

(t) := cl dec F (t), and examine which regularity properties of F (e.g., measurability, continuity) are preserved for G and G

. The results obtained are analogous to classical theorems on the convex hull of a multifunction.

2. Weakly measurable multifunctions

Throughout this section (T, Γ) is a measurable space, Y is a metric space,

and L

(Ω, X), 1 ≤ p < +∞, is defined for (Ω, Σ, m) with a finite measure

m. A multifunction F : T −→ P(Y ) is called weakly measurable if F satisfies

the condition F

(U ) ∈ Γ for each open set U ⊂ Y (cf. [2]). A countable

family V of measurable selections of F such that F (t) =cl{f (t) : f ∈ V } for

all t ∈ T is called a Castaing representation of F .

The following theorem characterizes weakly measurable multifunctions:

Theorem A ([2], Theorem 5.6). Let Y be a separable complete metric space, and F : T −→ P(Y ) a closed-valued multifunction. Then F is weakly measurable iff it has a Castaing representation.

Using this characterization we prove the following result:

Theorem 1. Suppose 1 ≤ p < +∞, L

(Ω, X) is separable, and F : T −→

P(L

(Ω, X) is a weakly measurable multifunction with closed values. Then the multifunctions G, G

: T −→ P(L

(Ω, X)) defined by G(t) := dec F (t), G

(t) := cl dec F (t), t ∈ T , are also weakly measurable.

In the proof of Theorem 1 we use two lemmas. For A, B ∈ Σ we denote by A ÷ B the symmetric difference of A and B.

Lemma 1. Suppose L

(Ω, X) is separable. Then there exists a countable family Π ⊂ Σ such that for each ² > 0 and each A ∈ Σ there is B ∈ Π satisfying m(A ÷ B) < ².

P roof. Fix z ∈ X with kzk = 1 and consider the set Z = {χ

z : A ∈ Σ}.

Since L

(Ω, X) is separable, Z has a countable dense subset E. Let Π be a countable family of B ∈ Σ such that χ

z ∈ E, where we identify A

, A

with m(A

÷ A

) = 0. Hence, for each ² > 0 and A ∈ Σ there exists B ∈ Π such that m(A ÷ B) = kχ

z − χ

zk

< ².

Lemma 2. Suppose L

(Ω, X) is separable. Let A be a subset of L

(Ω, X), B a dense subset of A, and Π such as in Lemma 1. Then

V = {χ

y

+ χ

y

+ . . . + χ

k−1\S_k−2

i=1 Bi

y

+ χ

i=1Bi

y

: B

, . . . , B

∈ Π, y

, . . . , y

∈ B, k ∈ N}

is a dense subset of cl dec A.

P roof. We shall show that for each z ∈ dec A and each ² > 0 there exists v ∈ V such that kz − vk

< ². Fix z ∈ dec A and ² > 0. The function z has a representation z = P

i=1

χ

z

where k ∈ N, z

∈ A, A

∈ Σ, i = 1, . . . , k, S

i=1

A

= Ω, A

∩ A

= ∅ for i 6= j. Since B is dense in A, for each

δ > 0 there exist y

, . . . , y

∈ B such that kz

− y

k

< δ, i = 1, . . . , k. Let

Π ⊂ Σ be such as in Lemma 1. For each η > 0 and each i = 1, . . . , k − 1 there is B

∈ Π satisfying m(A

÷ B

) < η. Denote

C

= B

, C

= B

\ B

, . . . , C

= B

\

k−2

[

i=1

B

, C

= Ω \

k−1

[

i=1

B

and v = P

i=1

χ

y

. By our construction, v ∈ V. We shall prove that for δ and η sufficiently small, kz − vk

< ². We have

kz − vk

= k X

(χ

z

− χ

y

k

≤ X

i=1

kχ

z

− χ

y

k

.

For a fixed i ∈ {1, 2, . . . , k}, kχ

z

− χ

y

k

=

Z

Ω

kχ

z

− χ

y

k

dm = Z

Ai∩Ci

kz

− y

k

dm

+ Z

Ai\Ci

kz

k

dm + Z

Ci\Ai

ky

k

dm.

Observe that R

Ai∩Ci

kz

− y

k

< δ

. For i < k we have C

\ A

⊂ B

\ A

and A

\C

= (A

∩B

)∪. . .∪(A

∩B

)∪(A

\B

). Consequently, m(C

\A

) < η and m(A

\ C

) < iη. Since A

= Ω \ S

i=1

A

, we have C

\ A

⊂

k−1

[

i=1

A

\ B

, A

\ C

⊂

k−1

[

i=1

B

\ A

.

Hence, m(C

\ A

) ≤ (k − 1)η and m(A

\ C

) ≤ (k − 1)η. By the ab- solute continuity of integral, for each γ > 0 there is η > 0 such that R

Ai\Ci

kz

k

dm < γ, R

Ci\Ai

ky

k

dm < γ, i = 1, 2, . . . , k. Thus we obtain the estimation kz − vk

≤ k(δ

+ 2γ)

. If δ, γ and η are sufficiently small, then kz−vk

< ². From here it follows that dec A ⊂ cl V and, consequently, cl dec A ⊂ cl V. A converse inclusion is obvious.

P roof of T heorem 1. By Theorem A the multifunction F has a Cas-

taing representation V

. We shall construct a Castaing representation for

the multifunction G

. Let Π ⊂ Σ be such as in Lemma 1. Denote

V := ©

g : T −→ L

(Ω, X) | g(t) = χ

f

(t) + χ

f

(t) + . . . + χ

k−1\S_k−2

i=1Bi

f

(t) + χ

i=1Bi

f

(t), B

, . . . , B

∈ Π f

, . . . , f

∈ V

, k ∈ N} .

Let us note that the set V is countable and by Lemma 2, G

(t) = cl {g(t) : g ∈ V} for each t ∈ T . Next we shall show that each g ∈ V is measur- able. To this end, fix C ∈ Σ and consider the function S

: v −→ χ

v, v ∈ L

(Ω, X). Since S

is continuous, all functions S

◦ f, f ∈ V

, are measurable. For fixed sets C

, . . . , C

∈ Σ and functions f

, . . . , f

∈ V

the function g defined by g(t) := χ

f

(t) + . . . + χ

f

(t) is measurable too, as the sum of measurable functions S

◦ f

, i = 1, . . . , k. Thus V is a Castaing representation of G

and, by Theorem A, we conclude that G

is weakly measurable. The multifunction G is weakly measurable too, because G

(U) = G

(U) for every open set U in L

(Ω, X).

3. Lower and upper semicontinuous multifunctions In this section, (T, τ

), (Y, τ

) denote topological spaces. Recall that a multifunction F : T −→ P(Y ) is called lower semicontinuous if for every U ∈ τ

, F

(U ) ∈ τ

, and F is called upper semicontinuous if for every U ∈ τ

, F

(U ) ∈ τ

.

For lower semicontinuous multifunctions F : T −→ P(L

(Ω, X)) we have the following theorem:

Theorem 2. Let 1 ≤ p < +∞ and F : T −→ P(L

(Ω, X)) be a lower semicontinuous multifunction. Then the multifunctions G, G

: T −→

P(L

(Ω, X)) defined by G(t) := dec F (t), G

(t) := cl dec F (t), t ∈ T , are also lower semicontinuous.

P roof. At first we shall show that for each open ball B(z, ²) in L

(Ω, X), the set G

(B(z, ²)) is open in T . Let us observe that t ∈ G

(B(z, ²)) iff there exists y ∈ G(t) satisfying ky − zk

< ². Each y ∈ G(t) can be represented as y = P

i=1

χ

y

where n ∈ N, y

∈ F (t), A

∈ Σ, for

i = 1, . . . , n, S

i=1

A

= Ω, A

∩ A

= ∅ for i 6= j. For such a function y we have

ky − zk

= Z

Ω

k X

χ

y

− zk

dm = X

Z

Ai

ky

− zk

dm.

The inequality ky − zk

< ² is satisfied iff there exist positive numbers

²

, . . . , ²

such that P

i=1

²

< ²

and R

Ai

ky

− zk

dm < ²

for i = 1, . . . , n.

For a fixed n ∈ N and ² > 0 we introduce the family D

:=

n

(A

, . . . , A

, ²

, . . . , ²

) : A

∈ Σ, ²

> 0, i = 1, . . . , n, [

A

= Ω,

A

∩ A

= ∅ for i 6= j, ²

+ . . . + ²

< ²

o

. Let V

:= {t ∈ T : there exists y ∈ F (t) such that R

A

ky − zk

dm < ²

}.

For α = (A

, . . . , A

, ²

, . . . , ²

) ∈ D

denote X

= T

i=1

V

. Now it is not difficult to see that G

(B(z, ²)) = S

n∈N

S

α∈Dn

X

. For fixed A ∈ Σ, z ∈ L

(Ω, X) and ² > 0 we denote

U

= n

x ∈ L

(Ω, X) : Z

A

kx − zk

dm < ²

o

.

Let us observe that V

= F

(U

). Since the function x −→ R

A

kx − zk

dm is continuous, U

is open in L

(Ω, X). Thus the sets V

and G

(B(z, ²)) are open in T . If U is an open set in L

(Ω, X), then

G

(U) = G

³ [

{B(z, ²) : z ∈ U, B(z, ²) ⊂ U}

´

= [ {G

(B(z, ²)) : z ∈ U, B(z, ²) ⊂ U}

is open in T . Consequently, the multifunctions G and G

are lower semi- continuous.

Remarks.

1. Our Theorem 2 also holds for L

(Ω, X); the above proof needs only

some obvious changes.

2. Theorem 2 is a generalization of Proposition 6.3 from [4], where the authors assumed that T is a σ-compact metric space. (We are grateful to the Referee for this information).

In the case of the upper semicontinuity there is no result similar to Theo- rem 2. Let us consider the following example, where T = X = R with the natural topology, Ω = [−1, 1], Σ is the σ-field of Lebesgue measurable sets and m is the Lebesgue measure.

Let F : R −→ P(L

([−1, 1], R)) be given as F (t) = {f

, f

}, where f

(u) = t, f

(u) = 0, u ∈ [−1, 1], t ∈ R. It is easy to show that F is upper semicontinuous.

We shall prove that G(t) = dec F (t), t ∈ R, is not upper semicontinuous.

Let (A

) be the sequence of subsets of [0, 1] defined by

A

= [0, 1], A

=

· 0, 1

2

¸

, . . . , A

=

"

0 ∗ µ 1

2

¶

, 1 ∗

µ 1 2

¶

#

∪

"

2 ∗ µ 1

2

¶

, 3 ∗

µ 1 2

¶

#

∪ . . . ∪

"

³

2

− 2

´

∗ µ 1

2

¶

,

³

2

− 1

´

∗ µ 1

2

¶

# ,

k = 3, 4, . . . and consider the functions h

= (1 +

)χ

∈ L

([−1, 1], R), n ∈ N. Let us note that kh

− h

k

>

for k 6= l. Thus the set {h

, h

, . . .}

has no accumulation point. Consequently, it is closed in L

. The set U = L

([−1, 1], R) \ {h

, h

, . . .} is open and G(1) = {χ

: A ∈ Σ} ⊂ U, since h

6∈ G(1) for every n ∈ N. Thus t = 1 ∈ G

(U) but (1−², 1+²) 6⊂ G

(U) for every ² > 0. Indeed, for each n ∈ N, h

= (1 +

)χ

∈ G(1 +

).

4. H-continuous multifunctions

Let (T, τ ) be a topological space and (Y, d) a metric space. For A ⊂ Y we

denote B(A, ²) = {y ∈ A : d(y, A) < ²}. A multifunction F : T −→ P(Y )

is called H-upper (H-lower) semicontinuous at t

∈ T iff for every ² > 0

there exists a neighbourhood U of t

such that for every t ∈ U, F (t) ⊂

B(F (t

), ²) (respectively, F (t

) ⊂ B(F (t), ²)). A multifunction F : T −→

P(Y ) is H-continuous iff F is H-upper and H-lower semicontinuous at every t ∈ T .

The following example shows that also in the case of the H-continuity there is no result similar to Theorem 2. Let T = X = R with the natural topology, Ω = [−1, 1], Σ be the σ-field of Lebesgue measurable sets and m the Lebesgue measure. We define F : R −→ P(L

([−1, 1], R)) as

F (t) =

( clB(θ, |t|), t 6= 0 {θ}, t = 0 where θ denotes zero in L

([−1, 1], R). Then

dec F (t) = cl dec F (t) =

( L

([−1, 1], R), t 6= 0

{θ}, t = 0

It is easy to show that F is H-continuous but G(t) = dec F (t) and G

(t) = cl dec F (t) are not H-upper semicontinuous at t

= 0. Note that unlike the previous example, the multifunction F is not upper semicontinuous, except t = 0.

In this case we have only the following theorem.

Theorem 3. Let T be a topological space and let f

: T −→ L

(Ω, X), i = 1, 2, . . . , k, be continuous functions. Then the multifunction F : T −→

P(L

(Ω, X)) defined by F (t) = dec {f

(t), . . . , f

(t)} is H-continuous.

P roof. Fix t

∈ T and ² > 0. Since f

, . . . , f

are continuous, there is a neighbourhood U of t

such that for each t ∈ U, kf

(t) − f

(t

)k ≤

², i = 1, 2, . . . , k. We shall prove that for t ∈ U we have F (t) ⊂ B(F (t

), ²) and F (t

) ⊂ B(F (t), ²).

In order to show the first inclusion, we fix t ∈ U and y ∈ F (t). Then

y = X

i=1

χ

f

(t),

where A

, . . . , A

∈ Σ, i = 1, . . . , k, S

i=1

A

= Ω, A

∩ A

= ∅ for i 6= j.

Observe that

z = X

i=1

χ

f

(t

) ∈ F (t

),

and

ky − zk

≤ X

i=1

kχ

(f

(t) − f

(t

))k

≤ X

kf

(t) − f

(t

)k

≤ ².

Hence, F (t) ⊂ B(F (t

), ²).

Analogously we prove that for each z ∈ F (t

) there is y ∈ F (t) such that kz − yk

< ², i.e., F (t

) ⊂ B(F (t), ²). It completes the proof.

References

[1] F. Hiai and H. Umegaki, Integrals, conditional expectations, and martingales of multivalued functions, J. Multivariate Anal. 7 (1977), 149–182.

[2] C.J. Himmelberg, Measurable relations, Fund. Math. 87 (1975), 53–72.

[3] Cz. Olech, Decomposability as substitute for convexity, in: Multifunctions and Integrands (ed. G. Salinetti), Lecture Notes in Math. 1091, Springer Verlag (1984), 193–205.

[4] A.A. Tolstonogov and D.A. Tolstonogov, L

-continuous extreme selectors of multifunctions with decomposable values: Existence theorems, Set-Valued Anal.

4 (1996), 173–203.

Received 3 April 2002