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

**A****n d r z e j**** K****a 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 *

**measure is said to be non-atomic if for every set A e E such that p(A ) Ф**

*\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 *

**[3],**

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

**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 (/*

a* (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.*

**0 are such that the**

*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 )*

### =

*\*

*у*

*\(А*

*а*

*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. *

*a*

* B) = 0*

### the function

*q*

* 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, *

*q*

*)*

*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,

*q*

*) 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 *

**a**x,**. . . ,**

**am**### 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*

*j*

* 2*

*j*

*ъ ) \ < — \*

*p*

*-*

*z*

*\*

*i*

**г = 1 j = l**

*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 *

**a**

*B) ^ J*

*l*

**l^i**

*— 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

II
*U iU C i* *if h <*

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

II
*Om*

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

**г=1**

*\p\{A *

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

**Lemma** **2**.

*Let (*

**Z***, *

*q*

*) be a complete mstric space, *

**2***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) = *

**p k**}**e**

**2 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*

^{2}

*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

orollary* 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.

**W**

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

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

**I h W**

**^ ( A ) = ^{AKf-^Q)).**

### Of course

**у**### =

**y x -\-y2.**### Corollary 2.2 from [1] says that if

**Q**### is any linear subspace of

**B n,***f* is the density of

**у, y**## x,

^{y 2}### are defined as above, then the range P of

**у**### is the algebraic sum of the ranges P 1, P 2 of

**y x**### ,

**y 2**### respectively.

### Therefore P =

**y { P )**### = P } + P 2 =

**y x***(27) + *

**y 2{ Z) .**### Since the density of

**y x**### 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

**Q1**

**P x cz Q. P n W Ф 0**

**P x c Q**

**y 2**### 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 **

**P2nW**

**P 2nW,**

**P 2nW = b. Using once more the lemma from [5] we see that**

**y f 1**

### is continuous at

**b.**### We shall prove now

**P****roposition**** 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 — *

**b,***p* | ->

**b,***(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

**P2nW,**

**W**

**P n W**

**P x + b. The convexity of Px**

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

**W, p e int**

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

### To prove the continuity of

**y ~ l***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, *

**j***= 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)\< *

**e2,**

*h [ y f } (Pj )*

*j*

* 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)*

**= ****у****Г 1 { v - b) ^ y l l (b).*******

### Taking any set

**A e***y~l (Pj), *

**y x{ A )**### =

**p ) , y 2{ A ) — p ) ,**### so there are

### the sets

**B xe y x ^(p***—*

**b)**### and

**B 2e y f 1 (b)**### such that

**\yx\(A***a*

**B x)***<*

**ex**### and

*\y2\* * (A *

*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)) = \[*

*л*

*2\{А*

*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

**T****heorem**

*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

**L****iapunov****’****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**

**[1] E. D. B o lk e r, Л class of convex bodies, Trans. Amer. Math. Soe. 145 (1969), **

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.**