On the continuity oî a mapping inverse to a vector measure

Séria I : P R A C E M ATEM ATYCZN E X V I I I (1974)

An d r z e j Ka r a f i a t

(Krakôw)

On the continuity oî a mapping inverse to a vector measure

This paper has been inspired by the results of C. Olech [5] and E. D. Bolker [1]. I t has come into existence thanks to Professor Olech’s valuable directions and aid. I am deeply grateful for his help I have had while writing it.

Thç purpose of this paper is to prove the uniform continuity of a mapping inverse to a finite, non-atomic vector measure (i. e. the mapping which associates a family of all measurable sets of the measure equal to p to the point p) in the range of this measure (Theorem 2). We give the Liapunov’s theorem [4] as an application. This theorem reads:

1° The range of a finite vector measure is closed.

2° The range of a finite, non-atomic vector measure is convex.

If a finite vector measure is non-atomic, then this theorem is a simple conclusion of the Theorem 2.

Let X be an abstract space, E a u-field of measurable subsets of X , p : E-^- R n a finite, non-atomic vector measure defined on E. A vector measure is said to be non-atomic if for every set A e E such that p(A ) Ф .0 there is a measurable subset В c A such that 0 Ф p (B ) Ф p{A ). Let

\p I be a total variation of p defined for A e E by the formula

\^\{A) = sup J^ \ p (A t)\,

i

where we take the supremum over all decompositions of A into finite number of disjoint subsets A i . p is absolutely continuous with respect to \p\ and \p\ is a non-atomic measure on E ([2], p. 123, [6], p. 98). Radon- Nikodym theorem implies the existence of a measurable function/: X -> B n which takes values from the unit sphere of B n almost everywhere in relation to \p\ in X ([1], Theorem 2.6) such that for every set A e E

p(A ) = f f(x)d\p\(x).

A

We shall call the function/a density of y. We can assume that/: X -*• Sn~l.

Let us denote by P = y(Z ) = {y(A)\Ae 27} the range of y.

Proposition

1 ([1]). I f there is a point a e 8n~l and a set A e 27 such th atf(x ) = a on A , then fo r every measurable subset В c A, y {B ) = a\y\(B).

Then the range o f y restricted to A is a segment.

In the space Z with a metric d we are able to define, according to

the Hausdorff distance between sets E , F c- Z:

h(E , F ) = max(supinf d(a, b), sup inf d(a, b)) if E , F Ф 0 ,

aeE be F beF aeE

h (E , 0 ) = sup d(a, b) if E Ф 0 , h (0 , 0 ) = 0.

a,beZ

The distance between E and 0 does not depend on E .

Since /: X -> S n~l, we are able to approximate uniformly the density / by simple functions gk measurable with respect to \y\. Then the function

y k (A) = f gk (x)d\y\(x) is a non-atomic vector measure, the set P k À

= {yk(A )\ A eZ } is a polytope and an algebraic sum of segments. If

\f(®)-9k(®)\ < £k on X , then, for every set A e Z, \y{A) - y k{A)\ < ek \y\{A) In this way we have obtained the following theorem ([1]):

Theorem

1. F o r each fin ite, non-atomic vector measure y and fo r each s > 0 there is a fin ite, non-atomic vector measure y 0 such that its range is an algebraic sum o f some segments and dist(//(27), yo{Z)) < s, where

“dist” is a H au sdorff distance in R n.

Corollary

1. The closure o f y(Z ) is convex.

P ro o f. Let P n be polytopes which approximate y(Z ). If dist (/

(27), P n) 0, then dist (/л (27), P n) -» 0 and closed limit of convex sets is convex.

We formulate a simple conclusion of Corollary 1:

Proposition 2. Let V be the smallest affin e subset o f B n which contains

y(Z ). Then with respect to the topology induced in V in ty(Z ) Ф 0 .

R e m a r k 1. From now on we shall assume that B n ■= V, i.e. 031^(27) Ф 0 and we shall denote y(Z ) by P.

Proposition 3. I f a point р е P and a number ô >

0 are such that the ball K ( p , 2ô) = {qe B n\ \q — p\ ф 2(5} c- P and P k is an approximating closed and convex polytope such that dist(P, P k) ^ Ô, then K (p , ô) c P k.

P ro o f. P k is an intersection of some half-spaces Н г, . . . , H r. If P k cr 1 then K ( p , 2ô) c- P and dist(P, P k) < 6, thus K (p , ô) c H {, there­

fore K { p , <5) c P k .

Proceeding to consider properties of function inverse to a vector measure, let us denote for each pair of sets A, B e 27

q{A, В )

=

where Д is the symmetric difference. On space 270 of equivalence classes of measurable sets with respect to relation

A ^ В о \/u\ (A.

B) = 0

the function

is a metric function ([2], p. 168, [5]). A ~ В implies p{A )

= p {B ), therefore we can consider p as defined on 270.

Let us denote the space of closed subsets of UQ by 2£o and consider this space with a Hausdorff distance h. I t is easy to prove that p i (E0,

)

B n is a continuous function. Thus p~ l (p) is closed for each point p e P . Therefore the mapping p~ xi P *p -> {A e E 0\p(A) = p } e 2Z’° as inverse to a continuous function is an upper semicontinuous mapping (its graph is closed). We can consider р~ г as defined on P by setting р~ г(р) — 0 for p c P \ P .

We shall precede the main theorem by some auxiliary informations and lemmas. I t is a well-known fact ([2], p. 169) that the metric space (Г 0,

) is complete, therefore, accordingly to [3], (2Z°, h) is the complete metric space too.

Lemma

1. I f p i E

B n is a non-atomic vector measure such that the range o f this measure W = p (E ) is a sum o f some segments and W is contained in the ball 1Ц0, r), then fo r each we W, s > 0 such that the ball K (w , 3e) c W and fo r each p a ir o f points p, qe K (w , e )

h(p 1{ p ) ,p l {q) ) < ---- Ip - q

£

P ro o f. I t is sufficient to show that for each A e p 1(p) there is a set В е р (q) such that д{А , Б ) < —- \p — q\ as the proof in the opposite nr

e direction is analogous.

Let W be a sum of segments

and let the ends of oq be 0 and a{ . We denote = / ~ 1( ——I, where / is a density of p. are

\ Ш /

Ш

disjoint and X = (J X { . For a fixed set A e p~ 1(p) let A i = A c\ X i and

i —l

Ш

h = Ы (Ai). Then p = £ 1{а{ . t=i If we denote

n 2e

q = p - ( q - p ) - ]--- r?

\q-p\

then \p — q°\ = 2e and q°e W. If vx, . . . , v°m are such that q° = £ г\а^

i= 1

m

then q = X vi ai ? where г'=1

vi

\p-q\

2s - [ 0 , 1],

2 s

\p-q\

2 s

0 1 11 ""г II a,\ <

г = 1

т о п

I р - д

2 s

Е ы

\ 1 пг

2

2

ъ ) \ < — \

-

\

where ( , ) is a scalar product in B n and ely en the natural basis because

\(aly е{)\ + \(а2, e{)\+ ... + I (am, ег-)|< 2r for each i.

Now we should verify only that there is a set B e у г(д) such that

m

\.(i\{BriXi) = vi and \p\{A

B) ^ J

— vt\\ai\’ There are sets Cly ..

г=1 such that 1и(С{) = \?н — г{\а{ and

Let us wirte

Ci c X {\A i if h < *i, Ci C Ai if h > vi,

Ci = 0 if

U iU C i if h <

B { = A i\ C i if h > Vi, if

Om

I t is easily verifiable that the set B = U B { satisfies p (B ) = q and

г=1

\p\{A

B ) < — \p — q\ which completes the proof of the Lemma 1. nr E

Lemma 2.

Let (

,

) be a complete mstric space,

z a space o f closed subsets o f Z with H au sdorff distance h, T, T k : Z - ^ B n, h = 1, 2, ... continuous functions such that T k converges, uniformly to T on Z. I f the functions Tk 1 : Tk(Z) * p k -> {ze Z\Tk{z) =

satisfy the Lipschitz condition with the same constant M on some open set Dcz T(Z) n U Tk(Z), then the function

k= 1

T -1 : I) * p -> {ze Z\T(z) = p } e 2Z satisfies the Lipschitz condition with the

same constant M on D.

P r o o f. {T k} is a Cauchy sequence, i.e. for each z e Z and for Jc, i > k{e) I T k{z)-T i{z)\ < £, and for every p, qe В such that \p — q\< ô, if T l (p ), then ze T f 1(K (p , e)) c T f l {K (q, £ + d)).

The Lipschitz condition implies that there is a ye T f 1(q) such that o{y, z) < M (s + (5). Similarly for each y e T f 1(q) there is a ze T k 1(p) such that g(y, z) < M (e + ô) i.e . Ji[Tk 1(p), T ^ 1(q)) < M (s + ô).

Therefore, we can denote by T° a limit of functions T k \ Now we shall show the identity I 7-1 == T° on T (Z )n D and containing T°(p) a T~1(p)

= 0 on B \ T (Z ). The uniform convergence Tk 1 to T° and T k to T implies for each p e D and z eT °(p ), z = lim

fc, where zk e Tk l {p) and p = T k(zk) -> T (z) i. e. T°(p) cz T~] (p). However, for each p e B n T ( Z ) and for each y e T ~ 1{p), y e T k 1{Tk {y))-> T°{T {y)). Consequently, Т~г(р) c T°(p) and the proof of Lemma 2 is completed.

The vector measure у : 27 -> B n can be approximated uniformly by vector measures y k such that P k = yk (Z) are polytopes (Theorem 1).

Thus 1\ converges uniformly to P = у (P), therefore P k converges to P too. For each point p e in tP there is a number ô > 0 such that K ( p , 6<5) с: P then K ( p , 3ô) c P k for Je > Je (ô) (Proposition 3). Lemma 1 implies that the functions y k l : P k -> 2Z° on the ball K ( p , d) satisfy the Lipschitz condition with a constant — -, where r — <5 + sup{|p| | YIT p e P } for Je > Je (ô).

ô

Lemma 2 implies therefore that the function y~ l satisfies the Lipschitz condition on the ball K ( p , ô). Consequently, we have the following:

C

2. TJie function y -1 inverse to a fin ite, non-atomic vector measure y is continuous on int P.

Now we shall consider the problem of the continuity of y~l (defined on P) on the boundary dP of P. Let us fix any point p e dP. There are two possibilities:

1° p is an extreme point of P or

2° there is an extreme face 17 of P , Jc — dim T7 > 1 such that p e int W.

The Lemma from [5] says that if P is convex, then an extreme point of P belongs to P and y~l is continuous at it. If 1° is true, then the conti­

nuity of y~ l at p is a simple conclusion of Corollary 1. Therefore, we can consider only 2°. In accordance with [1], Corollary 3.3, P has a /c-dimen- sional extreme face V if and only if \y\(f~1(Q)j > 0, where / is a density of y and Q is a ^-dimensional linear subspace of B n paralell to V. B y Krein-Milman theorem and the Lemma from [5], it follows that an exist­

ence of fc-dimensional extreme face 17 of P implies the inequality

и г ‘ <е)) > 0, where Q is a Jc-dimensional linear subspace paralell

to T7, because 17 as compact and convex contains extreme points

which belong to P . Consequently, P has a ^-dimensional extreme

face contained in W and the required inequality И (/ _1(ф)) > 0 is satisfied.

Let us define two non-atomic vector measures by formulas:

I h W = ^ ( A ) = ^{AKf-^Q)).

Of course

=

Corollary 2.2 from [1] says that if

is any linear subspace of

f is the density of

x,

are defined as above, then the range P of

is the algebraic sum of the ranges P 1, P 2 of

,

respectively.

Therefore P =

= P } + P 2 =

(27) +

Since the density of

takes values from Q1 so P x cz Q. P n W Ф 0 and P x c Q thus the Lemma from [5] implies that there is a point be P 2nW. I t follows that P 2nW = Ъ , because the vector measure

is identically equal to 0 on the subsets of /-1 (Q), all extreme points of P2nW belong to P and, accordingly to [1], Corollary 3.3 cited above, there is a unique extreme point of P 2nW, therefore P 2nW = b. Using once more the lemma from [5] we see that

y f 1

is continuous at

We shall prove now

Proposition 4.

Following the assumed notation

(i) I f any sequence o f points {pf} a P converges to p, then fo r every decomposition pj — p) + p f , where p)e P { (i = 1, 2), p) -> p —

p | ->

(ii) p — be in tP j.

P r o o f, (i) For every convergent subsequence of p 2, its limit belongs to P2nW, therefore is equal to b, so p 2 -> b and p ) - > p — b. To show (ii) let us notice that all extreme points of W as extreme points of P belong to P ([5]), so they belong to P n W = P x + b. The convexity of Px (Corol­

lary 1) and Krein-Milman theorem imply that Px -j- b = W, p e int IF, i.e.

p — b e i n t P 1 which completes the proof of Proposition 4.

To prove the continuity of

at p it is sufficient to show that for each sequence of points P j-^ p ( p ,P j* P ) a limit of y~1{pf) exists and this is equal to y ^ ip )-

We want to show that for each e > 0 there is a <5 > 0 such that if IP j~ P \ < ô, then h(y~'(pj), y -1 (p)) < e. Let pj — p) + p] , where p ) e P x {i = 1, 2,

= 1, 2, ...) . By Proposition 4 p 2 -> b, therefore for each e > 0 there is a (5 > 0 such that \Pj—p \ < ô implies \p2 — b\ < e and \p) ~ ( p —b)\

< е ф д . The continuity of y f 1 at b and /q~] at (p — b) (Proposition 4 (ii) and Corollary 2) implies that if \Pj—p\ < ô, then h ( y f 1(p2), y f l {b)\<

h [ y f } (Pj )

tx71{P ~ b )) < £i, where ex and e2-> 0 when ô-> 0. The set y~J (Pj) is the sum of [ y ^ iP j' j^ y f1 (Pj)] over all decompositions of pj into р ) + р ) (because A e у~г (pf) о y x(A) = p ) , y2{A) = p 2) and y~} (p)

= уГ 1 { v - b) ^ y l l (b). *

Taking any set

y~l (Pj),

=

so there are

the sets

and

such that

and

\y2\ (A

B 2) < e2. If we denote В = [B } n f ] {Q )]v [В2\ ^ (Q)], we have

\f*\((A Д B )n f-\ Q )) = \p1\ (A & B )< e1 and H ((J. д B)\f~* (Q)) = \[

2\{А

В)

< e2, that is \/u\(A д В) < e1 + e2, because \/л\ is a measure.

The proof in the opposite direction is analogous. I t follows then that the function /U1 is continuous at p. The point p was any point of dP , therefore the function is continuous on dP. Since it is continuous on in tP (Corollary 2), then it is continuous on P, therefore uniformly continuous. We have obtained the following

Theorem

2. I f p ,:U - > P n is a fin ite, non-atomic vector measure, p { £ ) = P , then the function p,~l : P ~ > 2Z° is uniformly continuous on P.

Theorem 2 and the definition of the Hausdorff distance imply that for every point p e P p~1(p) Ф 0 . Consequently, P c [p e R n\p~~l {p) Ф & )

= P . Using once more Corollary 1, we get

Liapunov’s theorem [4] (when a vector measure is non-atomic).

The range o f a fin ite, non-atomic vector measure is convex and closed.

E e m a rk . 2. In Theorem 2 the assumption that a vector measure is non-atomic and finite cannot be omitted.

References

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p . 323-345.

[2] P. R. H alm o s, Measure theory, Van Nostrand, New York 1950.

[3] K. K u ra to w s k i, Topology, vol. 1, Acad. Press, New York & London 1966.

[4] A. A. L ia p u n o v , Su r les fonctions-vecteurs complètement additives, Izv. Akad.

Nauk SSSR Ser. Math. 4 (1940), p. 465-478.

[5] C. O lech, On the range of an unbounded vector-valued measure, Math. Systems Theory 2 (1968), p. 251-256.

[6] — Extremal solutions of a control system, J . Diff. Equations 2 (1966), p. 74-101.