A note on strong convergence for Pettis integrable functions (revision)

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1 note on trong convergence for Petti integrable function (reviion) Erik J. Balder, Mathematical Intitute, Univerity of Utrecht, Utrecht, Netherland nna Rita Sambucini, Dipartimento di Matematica, Univerità di Perugia, Italy February, 2003 Mailing addre: Mathematical Intitute, Univerity of Utrecht, Budapetlaan 6, P.O. Box , 3508 T Utrecht, the Netherland balder@math.uu.nl and matear1@unipg.it, fax: Thi work wa upported by G.N..M.P.. of C.N.R. and wa obtained when the firt author wa viiting Perugia 1

2 1 Introduction In a recent paper [1] mrani, Cataing and Valadier obtained a convergence reult for a equence of Petti integrable function that continue a erie of reult tarted by Olech [11], Tartar [12] and Viintin [13]. Thee reult are of the following type. Given a weakly convergent equence of integrable function, one formulate an extreme point condition (pointwie) for the limit function. Thi eliminate peritent ocillation, and force the convergence to the limit function to be trong a well (for intance, convergence in meaure). By the ue of Young meaure, Viintin reult were improved in [4] and [14] and extended to Bochner integrable function with value in a eparable Banach pace. In thi note it i hown that Theorem 2.4, the main reult of [1], can be approached in a manner that i very imilar to the method introduced in [4]. Thi lead to a coniderable improvement of that reult, and in a direction that i uggeted on p. 331 of [1]. However, we have not been able to olve the open quetion on p. 331 completely. 2 Preliminarie Let (, Σ, µ) a complete probability pace. Let E be a eparable Banach pace. By E we denote the topological dual of E. Occaionally, the weak topology σ(e, E ) will be referred to by adding the ymbol w in the mathematical formula, and, likewie, the ymbol will indicate the trong (i.e., norm) topology on E. Let B(E) be the Borel σ-algebra on E; oberve that B(E) := B(E ) = B(E w ). Let C b (E ) be the et of all bounded -continuou function on E and let P(E) be the et of all probability meaure on (E, B(E)). We denote by R(; E) the et of all Young meaure from (, Σ) into (E, B(E)), i.e., the et of all function δ : P(E) uch that ω δ(ω)(b) i Σ-meaurable for every B B(S). equence {δ n } n in R(; E) converge narrowly (with repect to the -topology on E) to another Young meaure δ R(; E) if lim inf n [ E ] g(ω, x)δ n (ω)(dx) µ(dω) [ E ] g(ω, x)δ(ω)(dx) µ(dω) (1) for every meaurable function g : E [0, + ] uch that g(ω, ) i -lower emicontinuou on E for every ω. We hall denote uch convergence by δ n = δ. Several equivalent tatement of δ n = δ can be given; ee [7, Theorem 4.7]. Oberve that to any meaurable function f : E there correpond a Young meaure ε f in R(; E), called the relaxation of f. Thi i given by ε f (ω)(b) := 1 B (f(ω)); i.e., ε f (ω) i the point meaure (alia Dirac meaure) at f(ω). The following well-known reult (e.g., ee [6, Propoition 4.16]) will be very important: Propoition 2.1 For a equence {f n } n of meaurable function f n : E and another meaurable function f : E the following are equivalent: (a) ε fn = ε f, (b) {f n } n converge in meaure to f. Proof. By Theorem 4.7(c) in [7], (1) remain valid if intead of requiring g there to have value in [0, + ], we require g to have value in [ 1, + ]. So (a) (b) follow in an elementary way by uing, for arbitrary ɛ > 0, the function g ɛ : E [ 1, + ] given by { 1 if x f(ω) ɛ, g ɛ (ω, x) := 0 otherwie. Finally, (b) (a) follow by obviou argument involving the extraction of an a.e. convergent ubequence from {f n } n and Fatou lemma. QED Following [2], where a completely equivalent definition wa introduced (ee Remark 3.4 in [7]), a equence {δ n } n in R(; E) i called -tight (or imply tight) if for every ɛ > 0 there exit a meaurable multifunction Γ ɛ : k(e) uch that up δ n (ω)(s \ Γ ɛ (ω))dµ ɛ. n 2

3 Here k(e) denote the et of all -compact et in E. lo, a equence {f n } n of meaurable function f n : E i aid to be -tight if and only if {ε fn } n, the correponding equence of relaxation, i -tight [2], i.e., if and only if for every ɛ > 0 there exit a meaurable multifunction Γ ɛ : k(e) uch that up µ({ω : f n (ω) Γ ɛ (ω)}) ɛ. n N Specializing to the preent context, the fundamental lower cloure theorem for Young meaure [7, Theorem 4.13], which i equivalent to Prohorov theorem for Young meaure that wa given in [2, 5], take the following form. Theorem 2.2 Let {f n } n be a equence of meaurable function f n : E. If {ε fn } n in R(; E) i -tight, then there exit a ubequence {f nj } nj and an aociated δ R(; E) uch that [ ] lim inf g(ω, f nj (ω))µ(dω) g(ω, x)δ (ω)(dx) µ(dω) j E for every meaurable function g : E (, + ] for which g(ω, ) i lower emicontinuou on E and for which lim up max(0, g(, f n ( ))dµ = 0 α n {g(,f n( )) α} = δ. More- (i.e., {max(0, g(, f n ( )))} n i uniformly integrable). In particular, thi mean ε fnj over, δ (ω)(-l {f n (ω)}) = 1 for a.e. ω in, where -L {f n (ω)} := p=1-cl {f n (ω) : n p} i the pointwie Kuratowki lime uperior of the equence {f n (ω)} n. Let pa denote the purely atomic part of (, Σ, µ). can alway be choen in uch a way that for ome mea- Propoition 2.3 In Theorem 2.2 {f nj } nj urable function f : pa E f nj (ω)) f (ω) 0 for a.e. ω in pa. Proof. We argue a in [3] (ee alo the proof of Theorem 3.7 of [2]). The purely atomic part pa conit of at mot countably many non-null et that are atom j. On each atom j all Σ-meaurable function from into E are contant a.e. Moreover, by Cataing repreentation uch contancy on atom extend to meaurable cloed-valued multifunction. Fix j and chooe ɛ = µ( j )/2 in the definition of -tightne for (ε fn ); then there exit a Γ ɛ : k(e) uch that µ(f n Γ ɛ ) ɛ for all n N. Let K j E be the a.e.-contant value of Γ ɛ on j ; thi i a -compact et. Then it follow that for every n the a.e.-contant value on j of f n belong to K j. Hence, by a tandard diagonal extraction procedure we obtain a preliminary ubequence of {f n } n that -converge a.e. on pa = j j to ome f : pa E. fter thi, one applie Theorem 2.2. QED We finih thee preparation with the following reult. Here Lwc(E) denote the collection of all convex, cloed et in E that are weakly locally compact and do not contain any line. lo, τ(e, E) tand for the Mackey topology on E. Recall from [8, III.32] that there alway exit a countable, τ(e, E)-dene ubet of E. Lemma 2.4 Let ν P(E) and C Lwc(E) be uch that ν(c) = 1. If a ext C and if for a τ(e, E)-dene equence {x i } i in E x i, a = x i, x ν(dx) for every i N, then ν i the point meaure at a. C 3

4 Proof. Suppoe that ν were not concentrated in {a}. Then there would certainly exit a cloed, bounded and convex ubet B of C uch that a B and γ := ν(b) > 0. lo, γ could not be equal to 1, or ele one would get a B by the Hahn-Banach theorem. So it would follow that γ lie in (0, 1). Now define two probability meaure on C by etting ν 1 := ν( B)/γ and ν 2 := ν( (C \ B))/(1 γ). Then ν = γν 1 + (1 γ)ν 2. Since B i bounded and E i eparable Banach, the barycenter a 1 of ν 1 exit in the form of the Bochner integral a 1 := B xν 1(dx). Clearly, a 1 B C by the Hahn-Banach theorem (ince B i cloed and convex) and a 1 atifie x, a 1 = x, x ν 1 (dx) = γ 1 x, x ν(dx) for every x E. It follow that C\B B x i, x ν 2 (dx) = (1 γ) 1 [ B C ] x i, x ν(dx) x i, x ν(dx) = B = (1 γ) 1 x i, a γa 1 for every i N. We denote by a 2 the element (a γa 1 )/(1 γ) in E. From the above we know x i, a 2 = x i, x ν 2 (dx) up x i, x for every i N. x C C\B In view of the fact that C i cloed, convex, weakly locally compact and contain no line, it follow from thi that a 2 belong to C, by [9] (ee alo [8, Lemma III.34]). Then a = γa 1 + (1 γ)a 2, with a 1, a 2 C. Thi contradict the hypothei that a i an extreme point of C. QED 3 From weak to trong convergence for Petti integrable function Recall that a calarly µ-integrable function f : E i Petti integrable if for every Σ there exit ν f () E uch that x, ν f () = x, f dµ for all x E. We denote by PE 1 (µ) the et of all Petti integrable function f : E. norm on P E 1 (µ) i defined by { } f P e := up x, f dµ : x 1. x E In [1] a equence {f n } n in PE 1 (µ) i called Petti uniformly integrable if for every ɛ > 0 there exit δ ɛ > 0 uch that for every Σ with µ() δ ɛ the following inequality hold: up 1 f n P e ɛ. n N We hall adhere to thi name, even though a more proper name for thi notion would have been Petti equi-abolute continuity or Petti uniform abolute continuity. lo, a equence {f n } n in PE 1 (µ) i aid to converge weakly to f P E 1 (µ) if x, f n dµ = x, f dµ for every Σ and x E. lim n Our main reult i a follow: Theorem 3.1 Let {f n } n be a equence in PE 1 (µ) uch that: 4

5 (a) {f n } n i -tight; (b) {f n } n i Petti uniformly integrable; (c) {f n } n converge weakly to f P 1 E (µ) with f(ω) ext(co cl -L{f n (ω)}) a.e. Suppoe alo that co cl -L {f n (ω)} belong to Lwc(E) a.e. Then lim n 0 f n f P e = 0. Thi extend Theorem 2.4, the main reult of [1] in everal way. Firt, our condition (c) give a partial anwer to the open quetion formulated on p. 331 of [1]: we are able to relax the correponding aumption in [1] that there be a multifunction Φ : Lwc(E) with {f n (ω)} n Φ(ω) and f(ω) ext (Φ(ω)) a.e. For under uch a condition it i immediate that co cl -L{f n (ω)} Φ(ω) a.e., and f(ω) co cl -L{f n (ω)} Φ(ω) i a conequence of the ame argument a thoe that will be ued in our proof. Second, notice that two other improvement have been made: (i) more demanding tightne aumption i made in [1]: the multifunction Γ ɛ in the definition of tightne are not only to have value in the collection of all convex et in k(e), but Γ ɛ i alo required to be Petti-integrable itelf, which involve uniform integrability. (ii) While we are able to expre our reult in term of the Kuratowki -lime uperior, the open quetion in [1] i only formulated in term of the w-lime uperior. Our proof will make ue of the following traightforward reult in [1], which i a verion of the Vitali-Lebegue Theorem in PE 1 (µ). Propoition 3.2 ([1, Propoition 2.1]) Suppoe that {f n } n i a Petti uniformly integrable equence in P 1 E (µ) converging in meaure to f P 1 E (µ). Then lim n 0 f n f P e = 0. Proof of Theorem 3.1. We follow the idea of [4]. Let α := lim up n f n f P e. It i enough to how α = 0. Elementarily, there exit a ubequence {f m } m uch that α = lim m f m f P e. Conider the Young meaure relaxation ε fn of the function f n. Then (a) i equivalent to aying that the equence {ε fn } n N i tight in R(; E); o of coure {ε fm } m i tight a well. Therefore, by Theorem 2.2, there exit a ubequence {f m } m of {f m } m and a Young meaure δ R(; E) uch that ε fm = δ. We ditinguih now between what happen on pa, the purely atomic part of (, Σ, µ), and on na := \ pa, which i the nonatomic part of (, Σ, µ). On pa we know from Propoition 2.3 that, for ome meaurable f : pa E, we can uppoe without lo of generality that {f m (ω)} m -converge to f (ω) for a.e. ω in pa By weak convergence of {f m } m to the function f, it follow immediately that f = f a.e. on pa. So actually {f m } m converge to f a.e. on pa. Next, we conider what happen on na. Let {x i } i be any countable τ(e, E)-dene ubet of E. Then for every Σ, na, we have that ( ) x i, f m dµ x i, x δ (ω)(dx) µ(dω) E by applying Theorem 2.2 (ue both g(ω, x) := 1 (ω) x i, x and g (ω, x) := 1 (ω) x i, x ). Note alo in thi connection that { x i, f m ( ) } m i uniformly integrable, a a conequence of condition (b) and the nonatomicity of na. To be more precie, (b) by itelf implie that { x i, f m ( ) } m i equi-abolutely continuou, and then Exercie II.5.5 of [10] guarantee it uniform integrability on the nonatomic pace na. On the other hand, from the hypothei we know that x i, f m dµ x i, f dµ. So for almot every ω na x i, f(ω) = E x i, x δ (ω)(dx) for every i N. Moreover, Theorem 2.2 alo give δ (ω)(cl -L {f n (ω)}) = 1 for a.e. ω na. By Lemma 2.4, applied to ν := δ (ω), C := co cl -L {f n (ω)} and a := f(ω), we obtain that δ (ω) = ε f(ω) for 5

6 every non-exceptional ω in na. Recall that on pa we aw earlier that {f m } m -converge to f a.e. By Propoition 2.1 it follow that {f n } n µ-converge to f. So Propoition 3.2 give f m f P e 0. Since α = lim m f m f P e, it follow that α = 0. QED cknowledgment The author wih to thank an anonymou referee for helpful remark. Reference [1]. mrani, C. Cataing and M. Valadier, Convergence in Petti norm under extreme point condition, Vietnam J. Math. 26 (1998) [2] E.J. Balder, general approach to lower emicontinuity and lower cloure in optimal control theory, SIM J. Control Optim. 22 (1984) [3] E.J. Balder, unifying note on Fatou lemma in everal dimenion, Math. Oper. Re. 9 (1984) [4] E.J. Balder, On weak convergence implying trong convergence in L 1 -pace, Bull. utral. Math. Soc. 33 (1986) [5] E.J. Balder, On Prohorov theorem for tranition probabilitie, Séminaire nal. Convexe Montpellier 19 (1989) [6] E.J. Balder, Lecture on Young meaure theory and it application in economic, Rend. It. Mat. Univ. Triete, School on Meaure Theory and Real nalyi - Grado (Italy) 1997, 31 (2000), Suppl. 1, [7] E.J. Balder, New fundamental of Young meaure convergence, Calculu of Variation and Optimal Control, Reearch Note in Mathematic, Vol. 411,. Ioffe, et al. (ed.), Chapman and Hall/CRC, Boca Raton, 2000, pp [8] C. Cataing and M. Valadier, Convex nalyi and Meaurable Multifunction, Lecture Note in Mathematic, Vol. 580, Springer-Verlag, [9] V. Klee and C. Olech, Characterization of a cla of convex et, Math. Scand. 20 (1967) [10] J. Neveu, Mathematical Foundation of the Calculu of Probability, Holden-Day, San Francico, [11] C. Olech, Extremal olution of a control ytem, J. Differential Equation 2 (1966) [12] L. Tartar, The appearance of ocillation in optimization problem, Nonclaical Continuum Mechanic, Durham 1986, R.J. Knop, et al. (ed.), London Mathematical Society Lecture Note, Vol. 122, Cambridge Univerity Pre, Cambridge, 1987, pp [13] M. Viintin, Strong convergence reult related to trict convexity, Comment. Partial Diff. Equation 9 (1984) [14] M. Valadier, Différent ca où, grâce à une propriété d extrémalité, une uite de fonction intégrable faiblement convergente converge fortement, Sém. nal. Convexe Montpellier 19 (1989) [15] M. Valadier, Young meaure, weak and trong convergence and the Viintin-Balder theorem, Set-Valued nal. 2 (1994),

arxiv: v1 [math.mg] 25 Aug 2011

arxiv: v1 [math.mg] 25 Aug 2011 ABSORBING ANGLES, STEINER MINIMAL TREES, AND ANTIPODALITY HORST MARTINI, KONRAD J. SWANEPOEL, AND P. OLOFF DE WET arxiv:08.5046v [math.mg] 25 Aug 20 Abtract. We give a new proof that a tar {op i : i =,...,