J. Comp. & Math. Sci. Vol.5 (1), 79-83 (2014)
Existence of Measurable Selectors in Pettis Integrable Multi Function S. K. Pandey, D. K. Singh, Pankaj Kumar and Mithlesh Kumar Govt. Vivekanand P.G. College, Maihar Distt-Satna, M.P., INDIA. Kendriya Vidyalaya, Bailey Road, Patna, INDIA. Kendriya Vidyalaya, Lekhapani, Tinsukia, Assam, INDIA. (Received on: February 14, 2014) ABSTRACT Kuratowski and Ryll-Nardzewski's9 theorem about the existence of measurable selectors for multifunction is one of the keystones for the study of setvalued integration. In this paper we study Pettis integrability for multi functions and we obtain a Kuratowski and Ryll-Nardzewski's type selection theorem. We show that any Pettis integrable multi function F:cwk(X) defined in a complete finite measure space (, , ) with values in the family cwk(X) of all non-empty convex weakly compact subsets of a general Banach space X always admits Pettis integrable (measurable) selectors and that, moreover for each A the Pettis integral
Fd
coincides with the closure of the set of
A
integrals over A of all Pettis integrable selectors of F. In our paper we generalize the result of the research paper of CASCALES. B, KADETS. V, ROD RIGUES. J.2. Keywords: Multi function, Banach space, Pettis integral, Pettis integrable selector, cwk denotes for convex weakly compact, SMSP denotes scalarly measurable selector property.
1. INTRODUCTION Set valued integration has its origin in the seminal papers by Aumann1 and
Debreu9 and has been a very useful tool in areas like optimization and mathematical economics. The set-valued Pettis integral theory which goes back to the monograph by
Journal of Computer and Mathematical Sciences Vol. 5, Issue 1, 28 February, 2014 Pages (1-122)
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S. K. Pandey, et al., J. Comp. & Math. Sci. Vol.5 (1), 79-83 (2014)
Castaing and Valadier4 has attracted recently the attention of several authors. All these studies deal with multi functions whose values are subsets of a Banach space X, that is always assumed to be separable. The main reason for this limitation on X relies on the fact that an integrable multi function should have integrable (measurable) selectors and the tool to find these measurable selectors has always been the well known selection theorem of Kuratowski and Ryll-Nardzewski [9] that only works when the range space is separable. In this paper Pettis integral theory for multi functions can be done without the restriction of separability on the range space. Throughout this paper (, , ) is a complete finite measure space. X is a Banach space. Bx is closed unit ball of X and X* stands for the topological dual of X. Given x* X* and x X we write either (x*, x) or x* (x) to denote the evaluations of x* at x. 2. DEFINITIONS 2.1 A function f: X is said to be scalarly measurable if for each x* X* the composition <x*, f> = x* of : R is measurable, f is said to be Pettis integrable if
2.2 Measurable selectors for multi function F:cwk(X) which satisfy one of the following mathematical properties (2.2.1) {w : F(w) M} for every closed half space MX (Equivalently F is Scalarly measurable) (2.2.2) {w : F(w) M} for every closed set MX. (2.2.3) {w : F(w) M} for every norm closed set MX when X is separable. 2.3 Let X be a separable Banach space and F:cwk(X) a multi function. The following conditions are equivalents(2.3.1) F is Pettis integrable (2.3.2) The family WF = {* (x*, F) : x* BX*} is uniformly integrable. (2.3.3) The family WF is made up of measurable functions and any Scalarly measurable Selector of F is Pettis integrable. In this case each A the integral
Fd
coincides with the set of integrals
A
over A of all Pettis integrable Selector of F.
(2.1.1) x* of is integrable for every x* X*
2.4 Let F: cwk(X) be a multi function such that *(x*, F) is integrable for every x*X*. The following statements are equivalent.
(2.1.2) for each A there is an element
(2.4.1) F is Pettis integrable
fdμ X A
such that <x*,
fdμ > =
A
x * o fdμ for every x* X*
A
(2.4.2) For
each
,
the
mapping
F
: X* R ,
A
x* δ * (x*, F)dμ
A
is (X*,x)- continuous.
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S. K. Pandey, et al., J. Comp. & Math. Sci. Vol.5 (1), 79-83 (2014)
2.5 Let F, G: cwk(X) be two multi functions such that F is Pettis integrable, G is scalarly measurable and for each x* X*, we have *(x*,G) ≤ *(x*, F)-a.e. Then G is Pettis integrable and
Gd Fd A
for
A
every A .
Let g : X be any selector of G. Clearly g is also a selector of F. Now we claim g is measurable. we observe that
scalarly
* ( x0* , Gd ) * ( x0* , G )d
* ( x0* , F )d * ( x0* , Fd )
2.6 Let F: cwk(X) be a Pettis integrable multi function. If f:X is a scalarly measurable selector of F, then f is Pettis integrable and
x0* , ( x0 ) ( * ( x0* , G)d * ( x0* , Gd )
Fd Fd
It follows that
A
for every A .
A
Gd {x } 0
2.7 Let F: cwk(X) be a scalarly measurable multifuncton. Fix x*0 X* and consider the multi function G:cwk(X) G(w) : = {x F(w) : x0* (x1) = *(x0, F(w))}. Then G is scalarly measurable. 3. MAIN RESULTS Theorem 3.1 Let F : cwk (X) be a Pettis integrable multi function. Then F admits a and Pettis integrable (measurable) selector. Proof: Let us consider the multifunction G:cwk(X) G(w) : = {x F(w) : x0* (x) = *(x0*, F(w))}. By 2.7, G is scalarly measurable. Since G(w) F(w) for every w and F is Pettis integrable By 2.5, G is Pettis integrable with
Gdμ Fdμ .
Ω
Ω
Given x0* X * , we have
* ( x * , G ) * ( x * , G ) and
(
*
Hence
( x * , G) d x * ( x0 ) * ( x* , G)d
* ( x * , G ) * ( x * , G )
a.e.
Therefore x* og = * (x*, G) a.e. and x* og is measurable. Since x* X* is arbitrary g is scalarly measurable. By 2.6, we conclude that g is Pettis integrable In this way g is measurable Pettis integrable selector. Hence we get the required result. Proved. Theorem 3.2: Suppose X has the SM SP. let F:cwk(X) be a scalarly measurable multi function such that every scalarly measurable selectors of F is Pettis integrable. Then F is Pettis integrable.
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S. K. Pandey, et al., J. Comp. & Math. Sci. Vol.5 (1), 79-83 (2014)
Proof: For any fixed A . The set IS F ( A) is closed and convex. Now we prove that IS F ( A) cwk (X). Fix x* X* and consider the multi function Gx* : cwk(X) Gx* (w) = {xF(w): x*(x) = * (x*, F(w))}. By 2.7 Gx* is scalarly measurable and X has the SMSP, there is a scalarly measurable selector gx* of Gx*. gx* is a selector of F and so it is Pettis integrable. Hence *(x*, F) = x* ogx* is integrable. By the definition
g
x*
d IS F ( A)
we claim that Sub {x * ( x) : x IS F ( A)} x * ( g x* d ) .
A
Indeed notice that for each Pettis integrable selector f of F, we have x* g x*d A
x
*
x * og x * d * ( x * , F ) d A A
of x * d x * (
A
fd ) A
*
{ x ( x ) : x IS F ( A )} Sub { x * ( x ) : x IS F ( A )}
Sup
x * g x *d A
IS F ( A) is weakly compact. *(x*, IS F ( A) = * ( x * , F ) d
A
F is Pettis integrable Proved.
REFERENCES 1. Aumann, R.J. Integrals of set valued functions, J. Math. Anal. Appl. 12, 1-12 (1965). 1. Bourgin, R.D. Geometric aspects of convex sets with the Rodon. Nikodym Property, Lectures notes in Mathematics, Vol. 993, Sprmiger-Verlag, Berln (1983). 2. Cascales, B, Kadets, V & Rodriguez. J. The Pettis integral for multi valued functions via single valued ones, J. Math. Anal. Appl. 332 No.1, 1-10 (2007). 3. Cascales, B, & Rodriguez, J. Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297, No. 2, 540-560, special issue dedicated to John Horvath, (2004). 4. Castaing. C & Valadier. M. Convex analysis and measurable multi functions, Springer Verlag Berlin, Lectures Notes in Mathematics, Vol. 580 (1977). 5. Coste. A: Proceedings of the conference topology and Measure-1, pp-55-74 (1978). 6. Edgar. G.A. Measurability in a Banach space, Indian Univ. Math. J. 26, No. 6, 663-677 (1977). 8. EI Amri. K & Hess. C. On the Pettis integral of closed valued multi functions set valued Anal, B (2000). 9. Hess. C. Set valued integration and set valued probability theory: an overview, Hand book of measure theory, Vol I, II North- Holl and Amsterdam, pp 617-673 (2002). 9. Kuratowski. K & Ryll-Nardzewski's. C. A integral theorem on selectors, Bull. Acad. Polon. Sci. Ser. Sci Math. Astronom. Phy. 13, 397-403 (1965).
Journal of Computer and Mathematical Sciences Vol. 5, Issue 1, 28 February, 2014 Pages (1-122)
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10. Raja. M: First Borel class sets in Banach spaces and the asymptotic-norming property, Irael J. Math., 138, 253-270 (2003). 11. Talagrand. M: Pettis integral and Measure theory, Mens. Amer. Math. Soc.
83
51, No. 307 (1984). 12. Ziat. H: On a characterization of Pettis integrable multi functions. Bull. Polish Acad. Sci. Math. 48, No.3, 227-230 (2000).
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