R O C Z N IK I P O L S K IE G O T O W A R Z Y S T W A M A TEM A TY C ZN EG O Séria I : P E A C E M A T EM A T Y C Z N E X X I I I (1983)
W. O
r l i c zand E . U
r b a n s k i(Poznan)
Total variation oî a set-valued measure
Summary. In this paper there are given necessary and sufficient conditions in order that the total variation v of set-valued measure Ж be of a finite measure.
0. Basic definitions and notation. Throughout this paper, 8 is an abstract space,
£% is a cr-field of subsets of 8,
X is a Banach space with norm || ||. - 0.1. D
e f i n i t i o n s:
(1) s/{X ) is the family of all non-empty subsets of X . (2) &(X) is the family of all bounded sets in s4{X ).
Let 1 be any index set; then
(3) f ( I ) = { j : j a I and j is finite}.
Let (An)n>! c i ( I ) ; then
(4) xn< A n iff for every n, xn e A n.
(5) (An) is (s.p.d.) iff A n is a pairwise disjoint sequence.
(6) A n = {œ : æ = ]? a>n for some (ccn) < (An) and the series xn is
1 *■
convergent to ж}.
For M : &t->sé(X) such that M (0) = {0},
(7) M is additive if M(A\jB) = M (A )+ M {B ) whenever A, В are disjoint.
(8) M is a set-valued measure on ffl iff for every (A n) s.p.d. whose union is A we have M(A) = £ M {A n).
1
(9) Ж is a strict set-valued measure on M iff Ж is a set-valued measure and additionally, for every жга, such that {xn) < [M {An)~\, the series
£ x n is unconditionally convergent where (An) is s.p.d.
(10) AT'' (E) = U {M (F) : F cz E , F e Щ . (11) AT is bounded-valued if AT: M-^08{X).
(12) AT is bounded if AT is bounded-valued.
(13) A set A o f M is an atom of AT if AT (А) Ф 0 and for every В cz A either ATv (В ) = {0} or AT" (A \ B ) = {0}.
(11) AT is non-atomie if it has no atoms.
(15) The total variation v of AT is
v(M, E) = sup{^||®J : (a>it Ef) is a finite sequence in I x f , (Ef) is a partition of E and (x{) < AI(Ei)\.
(16) r](AT,E) = sup {||o?||: ж e Ж "(Æ7)}.
For r: ^ со],
(17) AT < v if for every E e t](AT, E) < v(E).
(18) v is exhaustive or s-bounded if for every (E n) (s.p.d.) lim v{En) = 0 . П
(19) v is order continuous if, for every sequence (En) cz â%, E n\ Ф implies lim r(F n) = 0 .
П
In this paper we assume AI: (X) and AI(0) = {0}.
1. Conditions for a bounded set-valued measure AT to be a strict set
valued measure. I t is clear that a strict set-valued measure Ж is a set- valued measure. '
Connections between the bounded set-valued measure and strict set-valued measure are following:
L
e m m a1.1 ([3], Theorem 5.11). I f AT is a strict set-valued measure, then AT is bounded-valued besides except at most a finite number o f atoms of AT.
C
o r o l l a r y1.2. I f AT is a strict set-valued measure and AT is non-atomie, then AT is bounded.
T
h e o r e m1.3. I f AT is a set-valued measure such that AT is exhaustiveу then AI is a strict set-valued measure.
P ro o f. Let (An) be (s.p.d.); we show.that for every xn < AT(An), ]? x n
«>1 satisfies the Cauchy condition.
Otherwise there is an e.> 0 such that for every J Q e f(N ) there exists a К e f ( N \ J 0) with || > e.
n e K
I t is seen that there is a finite sequence (K m) cz f{N ) such that К ю n
г\Кг — 0 if m Ф l and
But (ixn) < 31 (An) ; hence
where
Thus
У e Л Щ А п) = Ж( U Л ) = M [F m),
n e K m n e K m n e^ m
E m = U A n> (F m) is s.p.d.
ne^m
yW m )> || J £ æn\\>e -
This contradicts the assumption that rj(31, •) is exhaustive.
C
o r o l l a r y1.4. I f for a set-valued measure 31 there exists a finite non
negative measure v on Si such that M < v, then 31 is a strict set-valued measure.
P ro o f. r]{M, E) v(E) implies that tj is exhaustive. Kow, 1.4 follows from 1.3.
In the case where dim X < oo, Z. Artstein [1] gives for every ill a con- ' struction of a finite non-negative measure y on f such that for every E
eM and x
eM (E) we have |ja?|| < v(E) < oo. The latter condition is equivalent to M < v.
2. Condition for a total variation of 31 to be a finite measure.
T heorem 2.1. I f 31 is a strict set-valued measure, then the total variation v of 31 is a measure on M.
P ro o f. See [5].
P
r o p o s it io n2.2. I f M < v, then the total variation v< ,v.
P ro o f. For every E e Si and
x e31(E) we have, by the definition of v(M ,E ), v(3I, E) ^ \\x\\, and hence 3 I< iv (3 I, •).
Eow, let v be a measure on Si such that 31 < v. Then for every finite partition of E and xi e 31 (Ef) we have
n n n
У M l < У У(Щ = V (U Щ = v[E) ;
ttl fXi г=1
hence
v(3I, E) < v(E) .
E e m ark . If the total variation v(3I, •) of 31 is a finite measure on Si, then it is easily observed that for every (E n) (s.p.d.) and for every (xn)
< 31 (En) we have У \\xn\\ < oo, and so it is a necessary condition that
the total variation v to be a finite measure.
T
h e o r e m2.3. I f fo r every sequence (En) of pairwise disjoint elements o f and fo r every (xn) < M (En) we have 2 \\xn\\ < oo, then the total variation
1 v(M , •) is exhaustive.
P ro o f. Let (En) be (s.p.d.). Suppose that v{M, E n) does not converge to 0. Then there exists a e0 > 0 such that for every n e N we have v(M, E n) > sQ. Hence, by the definition of the total variation v(M, •) of M, there exists a finite partition (E ni) of E n and (xni) < M (E ni) such that
v (M ,E n) < $e0 + £ Kill- Thus
(*) i ; i w > K > o .
Now, the sequence (F n) = (Eu , E 12, . .., Е щ , E 21, . .., Е щ , ...) is disjoint and (xn) = (®u , x12, . .. , хщ , x21, . .. , хщ , . . . ) < (Fn) but, by (*), 2
i kii = oo, which is a contradiction.
P
r o p o s it io n2.4. I f M is additive and bounded-valued, then the total variation v(M, •) is finite fo r every atom A o f M.
P ro o f. If A is an atom of M, then there exists a X 0 e s#(X) (see [3]) such that for every B a A, B e M
Щ В ) = {0} or M { B ) = X „J hence
v{M, A) — sup{||a?||: x e X 0} < oo,
because M is bounded-valued.
щтттмянт *
T
h e o r e m2.5. The total variation v(M, •) of M is a finite measure on i f f M is a bounded-valued and strict set-valued measure on S& such that for every sequence {En) f o f pairwise disjoint elements o f M and fo r every (xn)
< M (E n), we have ^||a?J| < °°-
P ro o f. Part “only if” is evident.
“I f ”: By Theorem 2.1, the total variation v (M, •) is a measure on and by 2.3 it is exhaustive. This implies that v(M, •) is order continuous.
Now from 5.5 of [2], there is a set A 0 eM such that v(M, -A0) and v(M, A) = 0 or v(M, A) = oo for each A eM such that i n i 0 = 0 . Therefore v(M, B) = oo implies В Ф A 0 and
В = B \ A 0 и B n A 0.
Thus,
v(M, B) = v (M , B \ A 0) + v { M ,B n A 0).
Now we have
v{M, B\-A0) = oo,
and B \ A 0 is an atom of v(M, •); this implies that B \ A 0 is an atom of M.
This contradicts Proposition 2.4.
References
[1] Z. A r ts te in , Set-valued measures, Trans. Amer. Math.. Soc. 165 (1972), 103-125.
[2] L. D re v n o w s k i, Topological rings of set, continuous set functions, integrations I I , Bull. Acad. Polon. Sci., Sér. Sci. Math., Astr. et Phys. 20 (1972), 277-286.
[3] —, Additive and countably additive correspondences, Comment. Math. 19 (1976), 25-54.
[4] D. S ch m e id le r, On set correspondences into uniformly convex Banach spaces, Proc. Amer. Math. Soc. 34 (1972), 97-101.
[5] R. U rb a n s k i, On set-valued measures and set correspondences, Fünctiones et Apro- ximatio 3 (1976), 259-269.
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