On measurable multivalued mappings

4.

On Measurable Multivalued Mappings

We shall note some of the basic properties of measurable multivalued mappings without striving for maximal generality. Let R be equipped with Lebesgue measure ~, and let A ~ R

be a measurable subset.

1.4.1. Definition. An m-mapping F:A-+/((~) is called measurable if for any open the set F$1(V) [or FZI(V)] is measurable. An equivalent condition is as follows: for any closed W c R n is measurable.

VcR n

the set F$1(W) [or YlZ(W)]

It is obvious that upper (or lower) semicontinuous m-mappings are measurable. !.4.2. Definition. A single-valued mapping f:A-+R ~ is called a measurable section of an m - m a p p i n g F : A - + K ( R ~ ) i f f is measurable and ~(t)~F(t) for almost all t6A. The next assertion g i v e s a number of equivalent definitions of a measurable multivalued mapping. 1.4.3.

THEOREM.

For an m-mapping F:A-+K(R")the following properties are equivalent:

a) F is measurable; b) for any point x~R" with rational coordinates the function ~ : A - + R ~, ~x(t)=p(x, is measurable (p is the metric in R~);

F(t))

c) there exists a countable set {fm}%=x of measurable sections of F such that

F (t)= U

(t)}

m~l

for almost every t6A; d) for any 6 > 0 there exists a closed set A s i a tinuous.

such that ~ ( A \ ~ 8 ) ~ 6 and F I ~

is con-

Applying this theorem to operations on m-mappings, it can be shown that as a result of the intersection or union of a countable number of measurable multivalued mappings we obtain a measurable m-mapping (in the latter case it is assumed that for each value of the argument the union of the images is contained in a compact set). The operations of the sum of measurable m-mappings, multiplication of a measurable m-mapping by a measurable function, and convex closure of a measurable mapping (see Sec. 2) also lead to measurable m-mappings. 1.1.4. Definition. The integral of a multivalued mapping F: A-+K(R ~) is the set of all integrals of summable sections of F, i.e.,

SF(t)dt=(! f(t)dt[fEL~(A), f(t)GF(t~ . . tCA). The next assertion, which is essentially equivalent to the familiar theorem of Lyapunov on vector measures, is one of the basic properties of the multivalued integral. 1.4.5.

THEOREM.

The integral

S F(t)dt is a convex set.

If F : A - + K ( R n) is measurable

and

[1F (t)1[-----sup {l] f 111fCF (t)} < z (t) for almost all

tGA,

where zEA-+R is a summable function, then

A

A

and is a closed set. 1.4.6.

Definition.

L e t l A ~ R be a compact set, and let E be a Banach space.

An m-map-

ping F:AXE--~K(R")satisfies the Caratheodory condition if a) for all fixed x6E the m-mapping F(., x ): A - + R ~ is measurable; b) for almost all fixed t~Athe m-mapping F(t,'.):E-->-R ~ is upper semicontinuous. An m-mapping F satisfies tke strong Caratheodory conditions if it satisfies condition a) and the condition 2782

b l) for almost all fixed t~A the m-mapping

F(t, .) :E-~R"

is continuous.

For m-mappings satisfying the strong Caratheodory conditions the following analogue of the Skorets-Dragoni property holds [66]. 1.4.7. THEOREM. Suppose an m-mapping F : & X R m - ~ K ( R ~) satisfies the strong Caratheodory conditions. Then for any 6 > 0 there exists a compact set A~cA, such that ~ ( A ~ A o ) < ~ and F]~• is continuous. We note that the corresponding property for m-mappings satisfying the usual (not strong) Caratheodory conditions does not hold (see the counterexample in [34]). The next assertion is a generalization of a result important in the theory of controllable systems which was proved in its original form by A. F. Fillippov and is known as "the implicit function lemma of A. F. Fillippov." 1.4.8. THEOREM. Suppose an m-mapping F: AXRm-~/((R ~) satisfies the strong Caratheodory conditions and U:A-~/((R TM)is measurable. Let g:A-+R be a measurable mapping such that g(06f(t U(0) for almost all t6A. Then there exists a measurable section u :A-~R ~ of the mmapping U such that g(t)6F(t,u(t)) for almost all t~A. TM

1.4.9. Definition. Let F : A X E - ~ K ( R ~) be some m-mappingo The operator PF assigning to each m-mapping Q:A + K(E) an m-mapping ~:A-~P(R~} according to the rule

O(t) =F(t, Q(t)), is called the composition operator generated by F. 1.4.10. THEORF2.1. If an m-mapping F:AXR=-+/f(R ~) satisfies the strong Caratheodory conditions, then F is compositionally measurable, i.e., the composition operator PF assigns measurable m-mappings. We note that although an m-mapping F satisfying the ordinary Caratheodory conditions is not, generally speaking, compositionally measurable (see the example in [34]) it possesses the following property. 1.4. Ii. THEOREM. Let E be a Banach space; let F:AX [~m-~I((R~)satisfy the Caratheodory conditions. Then for each measurable mapping q :A-~R'~ there exists a measurable m-mapping S :A-~K(R~), such that

S(t)~F(t, q(t)) for almost all tEA. Suppose F : A X R ~ - ~ K ( R ~) satisfies t h e Caratheodory conditions and the following additional condition: there exist summable functions ~, ~:A-+R such that

liE(t, x)II~<~z(t) +13(t)Ilxll for all (t, x)~b)
Definition.

The superposition operator generated by an m-mapping

F : A X I ~ ~-~

K ( R ~) is the m-mapping ~:C(A,Rm)-~P(LI(A,Rm), assigning to each mapping qCC(~,~)the set of all measurable sections of the m-mapping ~PF (q):A-~K(Rn), P p (q)----F(t, q (t)). 1.4.13. THEOREM. Suppose an m-mapping F has convex images, and let a:LI(A,~)-~E be a continuous linear operator in the Banach space E. Then the m-mapping ao~:C(~,.~m)-~Cv(~)is closed. 5.

Monotone and Accretive Multivalued Mapping s

Let E be a Banach space~ and let E* be its dual space. linear form defined by the condition

On E* • E we consider the bi-

<w, u>=w(u) for any

,w~E*I uEE.

1.5.1. Definition. A set G c E* • E is monotone if for any pairs (w l, u I) and (w2, u 2) belonging to G, <w2--Wx, u2--u1>~O holds. A monotone set which is not a subset of a broader monotone set is called maximally monotone. Let Y c E, T:Y + P(E*) be an m-mapping; we set Y = D(T).

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