Priority Programme 1962

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1 Prorty Programme 1962 The Weak Sequental Closure of Decomposable Sets n Lebesgue Spaces and ts Applcaton to Varatonal Geometry Patrck Mehltz, Gerd Wachsmuth Non-smooth and Complementarty-based Dstrbuted Parameter Systems: Smulaton and Herarchcal Optmzaton Preprnt Number SPP receved on Aprl 20, 2017

2 Edted by SPP1962 at Weerstrass Insttute for Appled Analyss and Stochastcs (WIAS) Lebnz Insttute n the Forschungsverbund Berln e.v. Mohrenstraße 39, Berln, Germany E-Mal: spp1962@was-berln.de World Wde Web:

3 The weak sequental closure of decomposable sets n Lebesgue spaces and ts applcaton to varatonal geometry Patrck Mehltz Gerd Wachsmuth Aprl 20, 2017 We provde a precse characterzaton of the weak sequental closure of nonempty, closed, decomposable sets n Lebesgue spaces. Therefore, we have to dstngush between the purely atomc and the nonatomc regme. In the latter case, we get a convexfcaton effect whch s related to Lyapunov s convexty theorem, and n the former case, the weak sequental closure equals the strong closure. The characterzaton of the weak sequental closure s utlzed to compute the lmtng normal cone to nonempty, closed, decomposable sets n Lebesgue spaces. Fnally, we gve an example for the possble nonclosedness of the lmtng normal cone n ths settng. Keywords: decomposable set, Lebesgue spaces, lmtng normal cone, measurablty, weak sequental closure MSC: 49J53, 28B05, 90C30 1 Introducton Pontwse defned sets n functon spaces are standard n order to determne the feasble regons of optmal control problems. Typcally, control functons are supposed to le wthn sets of the form C := {v L p (m; R q ) v(ω) C(ω) f.a.a. ω, Technsche Unverstät Bergakademe Freberg, Faculty of Mathematcs and Computer Scence, Freberg, Germany, mehltz@math.tu-freberg.de, dma/mtarbeter/patrck-mehltz Technsche Unverstät Chemntz, Faculty of Mathematcs, Professorshp Numercal Mathematcs (Partal Dfferental Equatons), Chemntz, Germany, gerd.wachsmuth@mathematk.tu-chemntz.de, dgl/people/wachsmuth/ 1

4 where (, Σ, m) s a complete, σ-fnte measure space and C : R q s a set-valued mappng whch possesses nonempty, closed mages and certan measurablty propertes. Note that we do not assume the convexty of the mages of C. The generalty of ths settng s necessary n order to deal, e.g., wth optmal control problems comprsng mxed control-state complementarty constrants, see Guo and Ye [2016], Mehltz and Wachsmuth [2016]. It s easy to check that the above set C s decomposable n the sense (A, v 1, v 2 ) Σ C C: χ A v 1 + (1 χ A ) v 2 C, where χ A denotes the characterstc functon of the set A. Conversely, t s well known, that every nonempty, closed, and decomposable set C L p (m; R q ) can be wrtten n the above form, see [Papageorgou and Kyrts-Yallourou, 2009, Theorem 6.4.6]. If the set C appears n the constrants of an optmzaton problem, then correspondng necessary optmalty condtons derved va varatonal analyss are lkely to contan dfferent tangent or normal cones to the set C at some reference pont. Ths motvated our study of the varatonal geometry of pontwse defned sets C n Lebesgue spaces, see Mehltz and Wachsmuth [2016]. In the latter paper, we obtaned explct formulas for the Fréchet, strong lmtng, and Clarke normal cone to the set C under mld assumptons. Moreover, we presented reasonable upper and lower bounds (w.r.t. set ncluson) of the lmtng normal cone whch led to the observaton that n the nonatomc stuaton, the lmtng normal cone to C s always dense n the correspondng Clarke normal cone. However, we dd not derve an explct formula for the lmtng normal cone and the weak tangent cone. Here, we want to close ths gap. It wll be shown that the lmtng normal cone to C equals the so-called weak sequental closure,.e. the set of all weak accumulaton ponts of sequences, of the correspondng strong lmtng normal cone, see Proposton 5.4. In contrast to the weak closure, whch s the closure w.r.t. the weak topology, the weak sequental closure does not need to be weakly sequentally closed. In fact, t mght not even be closed. Thus, t s possble that the lmtng normal cone to pontwse defned sets s not closed, see Example 5.6. Ths strange behavor of the lmtng normal cone was already demonstrated by an example n l 2 n [Mordukhovch, 2006, Example 1.7]. Thus, n order to characterze the lmtng normal cone to pontwse defned sets, t wll be necessary to clarfy how the weak sequental closure to a pontwse defned set n a Lebesgue space looks lke. Ths s the am of Sectons 3 and 4 of ths paper where we dstngush between the stuatons of a nonatomc and a purely atomc measure space, respectvely. If nonatomc measure spaces are under consderaton, we observe a convexfcaton effect when computng weak sequental closures, see Theorem 3.8, whch seems to be related to Lyapunov s convexty theorem, see [Aubn and Frankowska, 2009, Theorem 8.7.3]. Especally, we obtan the supplementary result that pontwse defned sets n Lebesgue spaces are weakly closed f and only f they are weakly sequentally closed under mld assumptons, see Corollary Ths s a remarkable property whch does not hold for arbtrary sets n nfnte-dmensonal Banach spaces. In the settng of purely atomc measure spaces, t s shown n Corollary 4.3 that the weak sequental closure and weak closure concde wth the norm closure of the underlyng set. 2

5 We organzed the paper as follows: In Secton 2, we state the basc notaton whch s used throughout the manuscrpt. Secton 3 s dedcated to the characterzaton of the weak sequental closure to pontwse defned sets n nonatomc measure spaces. We present some necessary and suffcent crtera for the closedness of the weak sequental closure. Some examples are ncluded to vsualze the theory. In Secton 4, we deal wth the computaton of the weak sequental closure of pontwse defned sets n purely atomc measure spaces. Fnally, n Secton 5, the lmtng normal cone and the weak tangent cone to a pontwse defned set n a Lebesgue space are characterzed usng our results on the weak sequental closure. By means of two examples, we show that nether the lmtng normal cone nor the weak tangent cone to a pontwse defned set need to be closed. 2 Notaton Basc notaton Let N, Z, R, and R n denote the natural numbers, the ntegers, the real numbers, and the set of all real vectors wth n N components, respectvely. Furthermore, we use Z := Z {, R := R {+, R + := {r R r > 0, and R + 0 := {r R r 0. The standard smplex n R n s defned va { n n := λ R n λ = 1, λ 1,..., λ n 0. The Eucldean nner product of two vectors x, y R n s represented by x y whle x := x x s the Eucldean norm of x. We use B := {x R n x 1 for the closed unt ball n R n. The mappng dst: R n 2 Rn R defned by x R n A R n : dst(x, A) := nf x y y A s called dstance functon. Let A X be a subset of a real Banach space X. The sets cl A, cl seq w A, and cl w A denote the norm closure of A, the weak sequental closure of A (.e. the set of all weak accumulaton ponts of sequences comng from A), and the weak closure of A (.e. the closure w.r.t. the weak topology). Due to [Meggnson, 1998, Proposton ], cl w A equals the set of all weak accumulaton ponts of nets comng from A. Note that we have cl A cl seq w A cl w A and these nclusons are, n general, strct. We use span A, bd A, cone A, conv A, and conv A to represent the smallest subspace of X contanng A, the boundary of A, the smallest (not necessarly convex) cone n X contanng A, the smallest convex set n X contanng A, and the smallest closed, convex set n X contanng A, respectvely. Furthermore, we defne the polar cone of A as A := {x X x A: x, x 0. Theren, X s the topologcal dual of X whle, : X X R denotes the assocated dual parng. Recall that the Banach space X s called reflexve whenever the canoncal embeddng X x, x X s surjectve. 3

6 Tangent and normal cones Let A X be a nonempty, closed subset of the real reflexve Banach space X and fx some x A. The nner (or adjacent) tangent cone, the tangent (or Boulgand) cone, and the weak tangent cone to A at x are defned by { TA( x) {t n R + such that t n 0: := d X, {d n X : d n d, x + t n d n A n N T A ( x) := { d X {t n R + {d n X : t n 0, d n d, x + t n d n A n N, T w A ( x) := { d X {t n R + {d n X : t n 0, d n d, x + t n d n A n N, respectvely, see e.g. [Aubn and Frankowska, 2009, Secton 4.1]. Note that the nner tangent cone and the tangent cone are always closed whle ths property s not nherent for the weak tangent cone. Obvously, we always have the nclusons T A ( x) T A( x) T w A ( x) by defnton. If A s convex, then all these cones concde wth cl cone(a { x). We call A dervable at x f T A ( x) = T A( x) holds. Thus, any convex set s dervable everywhere. One may check [Mehltz and Wachsmuth, 2016, Secton 2.2] for more general crtera ensurng the dervablty of a set. Amongst others, the fnte unon of dervable sets s dervable agan, see [Mehltz and Wachsmuth, 2016, Lemma 2.1]. Next, we ntroduce the Fréchet (or regular) normal cone, the strong (or norm) lmtng normal cone, the lmtng (or basc, Mordukhovch) normal cone, and the Clarke (or convexfed) normal cone to A at x (see Mordukhovch [2006], Geremew et al. [2009]) va N A ( x) := TA w ( x), { NA( x) S := η X {xn A {η n X : x n x, η n η, η n N A (x n ) n N, { N A ( x) := η X {xn A {η n X : x n x, η n η, η n N A (x n ) n N, N C A ( x) := conv N A ( x), respectvely. By defnton the Fréchet as well as the Clarke normal cone are closed and convex. A dagonal sequence argument reveals that the strong lmtng normal cone s closed as well. On the other hand, the lmtng normal cone does not need to be closed f X s nfnte dmensonal, see [Mordukhovch, 2006, Example 1.7]. We have the nclusons N A ( x) NA S( x) N A( x) NA C ( x) and, n general, these nclusons are strct. If the set A s convex, then all these cones concde wth (A { x). Clearly, f X s fnte dmensonal, then the strong lmtng normal cone and the lmtng normal cone concde. Measurablty, Lebesgue spaces, and decomposable sets Let (, Σ) be a measurable space and let Y be a separable metrc space. A set-valued mappng C : Y s called measurable f for any open set O Y, the premage C 1 (O) := {ω C(ω) O s measurable. If C s assumed to be closed-valued, then there are equvalent useful characterzatons of measurablty, see [Papageorgou and Kyrts-Yallourou, 2009, Theorem ]. Partcularly, the closed-valued mappng 4

7 C s measurable f and only f there exsts a sequence {c n : Y n N of measurable functons such that C(ω) = cl{c n (ω) n N s vald for every ω. The map C s called graph measurable f ts graph gph C := {(ω, y) Y y C(ω) s measurable w.r.t. the σ-algebra Σ B(Y ). Theren, B(Y ) denotes the Borelean σ- algebra nduced by the metrc space Y and Σ B(Y ) represents the smallest σ-algebra contanng the Cartesan product Σ B(Y ). Note that whenever C s a measurable setvalued mappng wth closed mages, then t s graph measurable, see [Papageorgou and Kyrts-Yallourou, 2009, Proposton ]. Now, let X be a separable metrc space as well. A mappng ϕ: X Y s called a Carathéodory functon f ϕ(, x): Y s measurable for all x X whle ϕ(ω, ): X Y s contnuous for all ω. By means of [Papageorgou and Kyrts-Yallourou, 2009, Theorem 6.2.6], any Carathéodory functon ϕ: X Y s Σ B(X)-measurable. Now, assume that (, Σ, m) s a complete, σ-fnte measure space. Recall that (, Σ, m) s nonatomc whenever for every set M Σ wth m(m) > 0, there s some set N Σ such that m(m) > m(n) > 0 s vald. On the other hand, (, Σ, m) s referred to as purely atomc f every set of postve measure n Σ s an atom,.e. for every M Σ wth m(m) > 0 and every N Σ wth N M, we obtan m(n) = 0 or m(n) = m(m). For any p [1, ] and any q N, we denote by L p (m; R q ) the usual Lebesgue space of (equvalence classes) of measurable functons mappng from to R q equpped wth the usual norm. For brevty, we set L p (m) := L p (m; R). Recall that for p [1, ), the dual space L p (m; R q ) s sometrc to L p (m; R q ) where p (1, ] such that 1/p + 1/p = 1 holds denotes the conjugate coeffcent of p. The correspondng dual parng n L p (m; R q ) s gven by v L p (m; R q ) η L p (m; R q ): η, v := v(ω) η(ω) dω. Here, we use dω nstead of dm(ω) snce the measure m s fxed throughout each secton of the paper. The measurable space (, Σ, m) s called separable f for any s [1, ), the Banach space L s (m) s separable. Note that an arbtrary doman R d equpped wth the correspondng Borelean σ-algebra and Lebesgue s measure forms a σ-fnte, nonatomc, and separable measure space, see e.g. [Adams and Fourner, 2003, Theorem 2.21]. Thus, ts formal completon, see [Bogachev, 2007, Secton 1.5], s a complete, σ-fnte, nonatomc, and separable measure space. Now, let (, Σ, m) be a complete, σ-fnte measure space agan. A set C L p (m; R q ) s called decomposable f for any trplet (A, v 1, v 2 ) Σ C C, we have the relaton χ A v 1 + (1 χ A )v 2 C. Theren, χ A L (m) denotes the characterstc functon of A whch has value 1 for all ω A and vanshes everywhere on \ A. Clearly, decomposablty s some knd of generalzed convexty. It dates back to Rockafellar, see Rockafellar [1968]. It s well known that a nonempty and closed set s decomposable f and only f there exsts a measurable set-valued mappng C : R q wth nonempty and closed mages such that C possesses the representaton C = {v L p (m; R q ) v(ω) C(ω) f.a.a. ω, 5

8 see [Papageorgou and Kyrts-Yallourou, 2009, Theorem 6.4.6]. One may check Ha and Umegak [1977] and [Papageorgou and Kyrts-Yallourou, 2009, Secton ] for propertes and calculus rules for decomposable sets. 3 Weak sequental closure n nonatomc measure spaces In ths secton, we are gong to characterze the weak sequental closure of arbtrary closed, decomposable sets n Lebesgue spaces defned by a nonatomc measure. To ths end, we defne an auxlary functon on R q n Secton 3.1. Wth the help of ths functon, we are able to characterze the weak sequental closure n Secton 3.2. In Secton 3.3, we nvestgate under whch crcumstances the weak sequental closure s closed. Fnally, we gve a couple of counterexamples n Secton An auxlary functon Let C R q be a closed set and let p (1, ) be fxed. We defne the auxlary functon γ C : R q R by { γ C (x) := nf λ v p λ, v C, λ v = x (1) for x conv C and by γ C (x) = + otherwse. Carathéodory s theorem mples that γ C (x) < + holds f and only f x conv C s vald. By convexty of v v p, we have γ C (v) = v p v C. (2) The followng lemmas show that γ C s lower semcontnuous and convex. Lemma 3.1. Let {x n n N R q be a sequence wth x n x for some x R q and lm nf n γ C (x n ) < +. Then, there exst λ, v C such that λ v = x and λ v p lm nf γ C(x n ). (3) n In partcular, γ C s lower semcontnuous and the nfmum n (1) s attaned for all x conv C. Proof. Frst, we extract a subsequence (wthout relabelng), such that γ C (x n ) s fnte for all n N and converges towards ts least accumulaton pont as n. By defnton of γ C (x n ), there exst sequences {λ (n) n N, {v (n) n N C wth x n = λ(n) v (n) and λ(n) v (n) p γ C (x n ) + 1/n for all n N. Snce s compact, we can further extract a subsequence (wthout relabelng) such that λ (n) λ. W.l.o.g., there s j {0,..., q + 1 such that λ > 0 for all {1,..., q + 1 wth j and λ = 0 for all {1,..., q + 1 wth > j. Ths mples v (n) p 1 λ (n) λ (n) v (n) p 1 λ (n) ( γ C (x n ) + 1 ) β < + for all j, n 6

9 where β R s an upper bound of the convergent sequences { 1/λ (n) (γ C (x n ) + 1 n ) n N, = 1,..., j. Hence, we can extract a subsequence such that v (n) v C as n for j. On the other hand, λ (n) v (n) = ( λ (n) ) 1 1/p ( (n) λ v (n) p) 1/p ( λ (n) ( ) 1 1/p for all > j. Now, we defne v C arbtrarly for > j. Ths yelds λ v = Smlarly, we have λ v p = j j λ v p + 0 lm n j λ v + 0 = lm n λ (n) j λ (n) v (n) + lm n =j+1 1/p λ (n) v (n) p) 0 = lm λ (n) n v (n) = lm x n = x. n v (n) p + lm n =j+1 λ (n) λ (n) v (n) v (n) p = lm n λ (n) v (n) p. Ths shows the exstence of λ and v satsfyng (3). By defnton of γ C, we even have γ C (x) λ v p lm nf γ C(x n ) n and ths yelds the lower semcontnuty of γ C. Choosng x n = x for all n N, we obtan that the nfmum n (1) s attaned. We use the above lemma n order to come up wth the followng observaton. Corollary 3.2. If the condton s satsfed, then conv C s closed. c > 0 x conv C : γ C (x) c ( x p + 1) (4) Proof. Suppose on the contrary that conv C s not closed. Then we fnd a sequence {x n conv C convergng to x / conv C. Notng that {x n s bounded, we obtan From Lemma 3.1, we easly see lm nf n γ C(x n ) lm nf n c( x n p + 1) < +. γ C (x) lm nf n γ C(x n ) < +. On the other hand, we have γ C (x) = + by defnton. Ths s a contradcton. 7

10 We note that the converse result does not hold, cf. Examples 3.17, 3.18 and Lemma 3.3. The functon γ C : R q R s convex. Proof. Choose x, y R q and κ (0, 1) arbtrarly. In the case x / conv C or y / conv C, we have γ C (x) = + or γ C (y) = +. Thus, there s nothng to show. Let us consder x, y conv C. By Lemma 3.1, we fnd λ x, λ y and v x, v y C satsfyng x = λx vx, y = λy vy as well as γ C (x) = λx vx p and γ C (y) = λy vy p. Now, we consder the followng lnear program n standard form: Mnmze µ v x p + ν v y p w.r.t. (µ, ν) R R s.t. µ + ν = 1, µ v x + ν v y = κx + (1 κ)y, µ, ν 0. (5) Obvously, (κλ x, (1 κ)λ y ) s a feasble pont of ths program, and due to the fact that ts feasble set s bounded, t possesses at least one optmal basc soluton ( µ, ν). Observe that (5) possesses q + 1 equalty constrants whch means that ( µ, ν) has at most q + 1 postve components. Ths shows γ C (κx + (1 κ)y) µ v x p + ν v y p κλ x v x p + (1 κ)λ y vy p = κγ C (x) + (1 κ)γ C (y). Ths yelds the convexty of γ C. Let us provde an auxlary result, whch wll be helpful later. We defne the functon ψ : R q() R R R q R va ψ(v 1,..., v, λ, α) := ( ) λ v, λ v p + α for all v 1,..., v R q, λ R, and α R. Obvously, ψ s contnuous and Lemma 3.1 mples ψ(c R + 0 ) = ep γ C := {(v, β) R q R γ C (v) β, (6) 8

11 where ep γ C s the epgraph of the functon γ C. Due to the lower semcontnuty of γ C, ts epgraph s closed. In the last result of ths secton, we apply the functon γ C to a decomposable set n a pontwse fashon. Lemma 3.4. Let (, Σ, m) be a complete and σ-fnte measure space and let p (1, ) be gven as above. Suppose that C : R q s measurable and closed-valued. Then, the functon Γ C : L p (m; R q ) R defned va Γ C (v) := γ C(ω) (v(ω)) dω R for all v L p (m; R q ) satsfyng v(ω) conv C(ω) for a.a. ω and Γ C (v) = + otherwse s well defned, convex, and lower semcontnuous. In partcular, t s weakly lower semcontnuous. Proof. Fx v L p (m; R q ) wth v(ω) conv C(ω) for a.a. ω. Then, the set-valued mappng Υ: R R q() defned by { Υ(ω) := (λ, w) C(ω) λ w = v(ω) for all ω s closed-valued, and measurable, see [Aubn and Frankowska, 2009, Theorem 8.2.9]. Furthermore, Υ(ω) s nonempty for a.a. ω. Snce we have { γ C(ω) (v(ω)) = nf λ w p (λ, w) Υ(ω) for almost all ω, the functon ω γ C(ω) (v(ω)) s measurable by means of [Aubn and Frankowska, 2009, Theorem ]. Together wth γ C(ω) (v(ω)) 0, ths shows that Γ C s well defned. The convexty of the functon Γ C s a smple consequence of the convexty of γ C(ω) for all ω, whch was provded n Lemma 3.3. Let us prove the lower semcontnuty of Γ C. Choose a sequence {v n n N of functons n L p (m; R q ) convergng to v such that v n (ω) conv C(ω) holds true for almost all ω for suffcently large n N (otherwse, there s nothng to show). Wthout relabelng, we extract a subsequence such that {Γ C (v n ) converges towards ts least accumulaton pont. From v n v n L p (m; R q ) we fnd a subsequence of {v n (wthout relabelng) whch converges pontwse almost everywhere on to v. Thus, we can apply Fatou s lemma and Lemma 3.1 n order to obtan lm nf Γ C(v n ) = lm Γ C(v n ) = lm γ C(ω) (v n (ω))dω n n n lm nf γ C(ω)(v n (ω))dω γ C(ω) (v(ω))dω = Γ C (v). n Ths completes the proof. 9

12 3.2 Characterzaton of the weak sequental closure In ths secton, we gve a characterzaton of the weak sequental closure of closed, decomposable sets n nonatomc measure spaces. Throughout ths secton, we use the followng assumpton on the measure space. Assumpton 3.5. We assume that (, Σ, m) s a σ-fnte, complete, separable, and nonatomc measure space. Moreover, let p (1, ) be gven. Frst, we recall the followng result whch s taken from [Papageorgou and Kyrts- Yallourou, 2009, Proposton ] and [Mehltz and Wachsmuth, 2016, Lemma 3.4]. Lemma 3.6. Suppose that (, Σ, m) satsfes Assumpton 3.5. Let C L p (m; R q ) be nonempty, closed, and decomposable. We denote by C : R q the assocated set-valued map. Then, we have and cl w C = { v L p (m; R q ) v(ω) conv C(ω) f.a.a. ω (7) conv C cl seq w C. The next lemma of ths secton demonstrates that a partton of unty 1 = λ (ω) such that λ: s measurable can be approxmated n the weak- topology of L (m) by sequences of characterstc functons. For ths result, t s essental that the measure m does not contan atoms. Lemma 3.7. Suppose that (, Σ, m) satsfes Assumpton 3.5. Let λ L (m; R ) wth λ(ω) for a.a. ω be gven. Then, there exsts a sequence {A (n) n N Σ, such that {A (n) s a partton of for all n N and χ A (n) λ n L (m) as n holds for all = 1,..., q + 1. Moreover, t s possble to choose {A (n) n N such that [ ι {1,..., q + 1: λ ι L 1 (m) = n N: m ( A (n) ) ι 2 ] λ ι (ω) dω. Proof. Frst, assume m() < +. Let e 1,..., e R denote the q + 1 unt vectors n R, set E := {e 1,..., e, and consder the assocated closed, decomposable set E := {ξ L p (m, R ) ξ(ω) E f.a.a. ω. From m() < +, E s nonempty. Thus, we obtan cl w E = {ξ L p (m, R ) ξ(ω) conv E f.a.a. ω = {ξ L p (m, R ) ξ(ω) f.a.a. ω, 10

13 .e. λ cl w E, from Lemma 3.6. Snce L p (m) s separable and reflexve whle E s bounded, we obtan λ cl seq w E from [Meggnson, 1998, Corollary ]. Thus, there s a sequence {ξ n n N E whch converges weakly to λ. Notng that {ξ n n N s bounded n L (m; R ), we already have ξ n λ n L (m; R ),.e. ξ n, λ n L (m) for all = 1,..., q + 1. By defnton of E, there exsts a partton {A (n) Σ of wth whch shows χ A (n) ξ n = e χ (n) A λ for all = 1,..., q + 1 as n whenever the underlyng measure space s fnte. For the proof n the case m() = +, we choose a partton { j j N Σ of nto sets of postve but fnte measure and repeat the above arguments on every segment j, j N. It remans to verfy the last asserton. By {A (n) n N Σ, we denote the sequence whch s gven by the frst asserton. Let ι {1,..., q + 1 be gven such that λ ι L 1 (m) holds. We denote by { j j N Σ a dsjont partton of nto sets of fnte measure. Snce χ j L 1 (m) s vald, we can fnd a subsequence ndexed by {n(j, k) k N whch satsfes and χ (n(j,k)) A ι j χ (n(j,k)) A ι j For = 1,..., q + 1 and k N, we set à (k) := j N dω λ ι dω as k j dω 2 λ ι dω k N. j j A (n(j,k)). Then, à (k) = {A (k) s a dsjont partton of for all k N as well. Let us consder the sequence {Ã(k) k N. For arbtrary v L 1 (m) and fxed {1,..., q + 1, we obtan vχã(k) dω = (v χ j ) χ (n(j,k)) A dω (v χ j ) λ dω = vλ dω j N j N as k,.e., χã(k) λ n L (m) as k. Note that we have used the domnated convergence theorem wth the summable domnatng sequence v χ j L 1 (m). On the other hand, m ( à (k) ) ι = m = j N j N j A (n(j,k)) ι χ (n(j,k)) A ι j = j N m ( j A (n(j,k)) ι dω 2 λ ι dω = 2 j N j ) λ ι dω 11

14 s vald. Thus, by replacng {A (n) n N by {Ã(k) k N and by repeatng the argumentaton for other ndces ι {1,..., q + 1 wth λ ι L 1 (m) yelds the second clam. Now, we are n the poston to prove the man result of ths secton. Theorem 3.8. Suppose that (, Σ, m) satsfes Assumpton 3.5. Let C L p (m; R q ) be closed and decomposable. We denote by C : R q the assocated set-valued map. Then, cl seq w C = {v L p (m; R q ) Γ C (v) < +. Proof. : Let v cl seq w C be gven. Then, there s a sequence {v n C satsfyng v n v n L p (m; R q ). Snce v n (ω) C(ω) for a.a. ω, we can use (2) and obtan Γ C (v n ) = γ C(ω) (v n (ω)) dω = v n (ω) p dω = v n p L p (m;r q ). Snce Γ C s weakly lower semcontnuous by Lemma 3.4, we fnd Γ C (v) lm nf n Γ C(v n ) = lm nf n v n p L p (m;r q ) < +. : Let v L p (m; R q ) wth Γ C (v) < + be gven. Then we have γ C(ω) (v(ω)) < + for a.a. ω. Let us ntroduce a set-valued mappng Φ: R R q() by { Φ(ω) := (λ, w) C(ω) λ w p = γ C(ω) (v(ω)) and λ w = v(ω) for all ω. Owng to Lemma 3.1, we fnd that the mages of Φ are not empty and t s straghtforward to check that they are closed. It s easy to see from [Aubn and Frankowska, 2009, Theorem 8.2.9] that Φ s measurable. By applyng the measurable selecton theorem, see [Aubn and Frankowska, 2009, Theorem 8.1.3], we fnd measurable functons λ: and v : R q, = 1,..., q + 1, whch satsfy v (ω) C(ω) for all = 1,..., q + 1, v(ω) = λ (ω) v (ω) for a.a. ω, and λ (ω) v (ω) p dω = γ C(ω) (v(ω)) dω < +. For {1,..., q + 1 and j Z, we defne { {ω 2 j v (ω) p < 2 j+1 f j >, (, j) = {ω v (ω) = 0 f j =. Furthermore, for a multndex J Z, we ntroduce J := (, J ). 12

15 Ths mples that { J J Z s a dsjont partton of. Now, we apply Lemma 3.7 on each of these J, J Z. Ths yelds sequences {A (J,n) n N such that {A (J,n) s a dsjont partton of J and χ (J,n) A λ J n L (m J ) as n. Moreover, we observe that λ s ntegrable on J f J >. In ths case, we addtonally have m(a (J,n) ) 2 J λ dω. Now, we set v (n) := v. Ths mples v (n) p L p (m;r q ) J Z J Z J Z J Z J Z = 2 1+p A (J,n) χ A (J,n) v p dω { m(a (J,n) ) 2 (J +1)p f J > 0 f J = { 2 2 p λ J 2 Jp dω f J > 0 f J = { 2 1+p λ J v p dω f J > 0 f J = λ v p dω = 2 1+p Γ C (v). Ths yelds the boundedness of {v (n) n N n L p (m; R q ), and v (n) C follows by constructon. It remans to show v (n) v n L p (m; R q ). Let J Z and {1,..., be gven. Frst, we consder the case J >. In ths case, χ (J,n) A v s bounded n L (m; R q ) and by Lemma 3.7 t converges weakly- towards χ J λ v n L (m; R q ) as n. By densty of L 1 (m) L p (m) n L p (m) and the boundedness of χ (J,n) A v n L p (m; R q ), ths v χ J λ v n L p (m; R q ) as n. yelds χ (J,n) A Now, we consder the case J =. Ths yelds v J χ A (J,n) v χ J λ v n L p (m; R q ) as n. = 0. Hence, we obvously have Ths yelds v (n), g v, g for all g L p (m; R q ) whch are supported on the unon of a fnte subset of { J J Z. Such functons are dense n L p (m; R q ) and together wth the boundedness of v (n) n L p (m; R q ) ths shows the clam. Let C L p (m; R q ) be nonempty, closed, and decomposable, let C : R q be the assocated measurable set-valued mappng, and let Assumpton 3.5 be vald. We want 13

16 to compare the above result for the weak sequental closure of C wth the expresson (7) for ts weak closure. By defnton of Γ C and the representaton of cl w C presented n Lemma 3.6, we obtan the nclusons cl seq w C {v L p (m; R q ) v(ω) conv C(ω) f.a.a. ω cl w C. (8) By means of an example, we wll see later that these nclusons can be strct at the same tme. In the upcomng theorem, we wll demonstrate that the weak closure of C can be obtaned by takng the norm closure of ts weak sequental closure. Theorem 3.9. Suppose that (, Σ, m) satsfes Assumpton 3.5. Let C L p (m; R q ) be decomposable. Then we have cl cl seq w C = cl w C. Proof. If C s empty, there s nothng to prove. In the case that C s closed and not empty, we nvoke Lemma 3.6 to obtan conv C cl seq w C cl w C. Takng the closure yelds conv C cl cl seq w C cl cl w C = cl w C. Now, observe that the set on the left s a closed as well as convex superset of C. Ths yelds cl w C conv C cl cl seq w C cl w C whch completes the proof n case that C s closed. It remans to consder the case that C s not closed. By applyng the frst part of the proof to cl C, whch s decomposable, we obtan cl cl seq w cl C = cl w cl C. By a dagonal sequence argument, we can show cl seq w cl C = cl seq w C and cl w cl C = cl w C s mmedate. Ths shows the clam n the case that C s not closed. As a corollary, we obtan an nterestng property of decomposable sets. Corollary Suppose that (, Σ, m) satsfes Assumpton 3.5. Let C L p (m; R q ) be decomposable. Then, C s weakly closed f and only f t s weakly sequentally closed. The above property of decomposable sets s very remarkable, snce there are examples for sets whch are weakly sequentally closed, but not weakly closed. For example, t can be checked that the set { n e n n N H, where {e n n N s an orthonormal bass n the Hlbert space H, s weakly sequentally closed, but 0 belongs to ts weak closure. 3.3 Closedness of the weak sequental closure In ths secton, we study necessary and suffcent condtons for the closedness of the weak sequental closure. The man motvaton for ths nvestgaton s that Theorem 3.9 mples cl seq w C = cl w C n the case that cl seq w C s closed, and we can use the more convenent formula (7) for the computaton of cl seq w C. Lemma Suppose that (, Σ, m) satsfes Assumpton 3.5. Let C L p (m; R q ) be nonempty, closed, and decomposable such that cl seq w C s closed. Let C : R q be the assocated set-valued mappng. Then, conv C(ω) s closed for almost every ω. 14

17 Proof. The closedness of cl seq w C mples cl seq w C { v L p (m; R q ) v(ω) conv C(ω) f.a.a. ω { v L p (m; R q ) v(ω) conv C(ω) f.a.a. ω = cl w C = cl seq w C, see Lemma 3.6 as well as Theorems 3.8 and 3.9. Thus, all these sets have to be equal. If we would already know that the set-valued mappngs ω conv C(ω) and ω conv C(ω) are graph measurable, we could apply [Papageorgou and Kyrts-Yallourou, 2009, Corollary 6.4.4] to get conv C(ω) = conv C(ω) for a.a. ω. Hence, t remans to show these graph measurabltes. From [Aubn and Frankowska, 2009, Theorem 8.2.2], we get that ω conv C(ω) s measurable. Consequently, t s graph measurable by [Papageorgou and Kyrts- Yallourou, 2009, Theorem ]. To show the graph measurablty of ω conv C(ω), we are gong to explot (6) and v conv C(ω) f and only f γ C(ω) (v) < +. To ths end, we ntroduce the Carathéodory functon ϕ: R q() R R R q R, defned va ( ) ϕ(ω, v 1,..., v, λ, α) = λ v, λ v p + α for all ω, v 1,..., v R q, λ R, and α R. By (6), we fnd ϕ ( ω, C(ω) R + ) 0 = ep γc(ω). Now, the set-valued map ω C(ω) R + 0 Frankowska, 2009, Theorem 8.2.8] mples that s measurable. Hence, [Aubn and ω cl ϕ ( ω, C(ω) R + ) 0 = cl ep γc(ω) = ep γ C(ω) s measurable and possesses nonempty, closed mages. For fxed K N, we defne a closed set M K R q R by M K := R q [0, K]. Now, we can nvoke [Aubn and Frankowska, 2009, Theorem 8.2.4] to see that ω M K ep γ C(ω) s measurable and closed-valued. Thus, [Aubn and Frankowska, 2009, Theorem 8.2.8] and [Papageorgou and Kyrts-Yallourou, 2009, Proposton ] can be used to obtan the graph measurablty of ω cl Proj(M K ep γ C(ω) ), where Proj: R q R R q s the projecton onto the frst q arguments. Usng the lower semcontnuty of γ C(ω) and the characterzaton Proj(M K ep γ C(ω) ) = {v R q γ C(ω) (v) K, we observe that the set Proj(M K ep γ C(ω) ) s already closed for a.a. ω. Hence, ω Proj(M K ep γ C(ω) ). 15

18 s graph measurable. Fnally, we note gph(ω conv C(ω)) = {(ω, v) v conv C(ω) = { (ω, v) 0 γ C(ω) (v) < + = { (ω, v) 0 γc(ω) (v) K K N = K N gph(ω Proj(M K ep γ C(ω) )). Hence, ω conv C(ω) s graph measurable. As already sad, [Papageorgou and Kyrts-Yallourou, 2009, Corollary 6.4.4] now mples that conv C(ω) = conv C(ω) for a.a. ω. In the case where the set-valued mappng whch characterzes the pontwse defned set C of nterest s constant, the stuaton s more comfortable. We study C = {v L p (m; R q ) v(ω) C f.a.a. ω (9) for a nonempty, closed set C R q and p (1, ). In ths settng, we fnd a condton whch s necessary and suffcent for the closedness of cl seq w C. However, we have to dstngush the cases where the underlyng measure space s fnte or not. Lemma Suppose that (, Σ, m) satsfes Assumpton 3.5. Let C be nonempty and defned as n (9) wth C R q beng nonempty and closed. We further suppose m() < +. Then, the closedness of cl seq w C s equvalent to condton (4). Proof. Suppose that (4) does not hold. Then there s a sequence {x n n N conv C wth γ C (x n ) > n ( x n p + 1). Ths mples that the sequence {γ C (x n )/( x n p + 1) n N s nonnegatve and does not belong to l. Hence, there s a nonnegatve sequence {α n n N l 1 wth γ C (x n ) α n x n p + 1 = +. We defne n=1 α n β n := x n p + 1, B := β n n=1 α n < +. Snce m s nonatomc, there s a countable partton { n n N of such that m( n ) = β n B 1 m() holds. Now, we consder the functon v := n=1 χ n x n. We have v p L p (m;r q ) = Γ C (v) = n=1 n=1 m( n ) x n p = m() B m( n ) γ C (x n ) = m() B n=1 n=1 β n x n p m() B n=1 α n < +, n=1 β n γ C (x n ) = m() B n=1 α n γ C (x n ) x n p + 1 = + 16

19 whch shows v cl w C \ cl seq w C. Ths, however, mples that cl seq w C s not closed, see Theorem 3.9. Ths fnshes the frst part of the proof. For the converse drecton of the proof, we assume that (4) holds. Let v cl w C be gven. Snce conv C s closed, see Corollary 3.2, we have v(ω) conv C for a.a. ω. Then, (4) mples γ C (v(ω)) c ( v(ω) p + 1). Hence, Γ C (v) = γ C (v(ω)) dω c ( v(ω) p + 1) dω = c ( v p L p (m;r q ) + m()) < + holds and ths shows v cl seq w C. Ths mples the closedness of cl seq w C, see Theorem 3.9. Lemma Suppose that (, Σ, m) satsfes Assumpton 3.5. Let C be nonempty and defned as n (9) wth C R q beng nonempty and closed. Furthermore, we assume that m() = + s vald. Then, the closedness of cl seq w C s equvalent to c > 0 x conv C : γ C (x) c x p. (10) Proof. Frst, let us assume that (10) s volated. We wll show that cl seq w C s not closed. There s a sequence {x n n N conv C wth γ C (x n ) > n x n p. Snce m() = +, and snce m s nonatomc, there s a countable sequence { n n N of parwse dsjont subsets of wth m( n ) = n 2 x n p. We set v := n=1 χ n x n. Then, we obtan v p L p (m;r q ) = x n p dω = m( n ) x n p = n=1 n n=1 n=1 Γ C (v) = γ C (x n ) dω m( n ) n x n p = n n=1 n=1 1 n 2 = π2 6, n=1 1 n = +. Hence, v cl w C \ cl seq w C. Invokng Theorem 3.9 shows that cl seq w C s not closed. Next, we assume that condton (10) holds. Choose an arbtrary sequence {v n n N C whch converges n L p (m; R q ) to some v L p (m; R q ). Explotng the lower semcontnuty of Γ C, see Lemma 3.4, and condton (10), we obtan Γ C ( v) lm nf Γ C(v n ) c lm nf v n (ω) p dω = c lm nf v n p n n n L p (m;r q ) < + cl seq w whch yelds v cl seq w C, see Theorem 3.8. Ths shows the closedness of cl seq w C and completes the proof. In the next lemma, we consder the case that C s a cone. Recall that B R q denotes the closed unt ball. Lemma Suppose that (, Σ, m) satsfes Assumpton 3.5. Let C R q be a nonempty and closed cone and consder the decomposable set C defned n (9). Then, C s closed f and only f there exsts some r > 0 such that the condton cl seq w s vald. (conv C) B conv(c rb) (11) 17

20 Proof. Frst, let us assume that there s some r > 0 such that (conv C) B conv(c rb) s vald. Choose some x (conv C) \ {0 arbtrarly. Then x/ x (conv C) B s vald and, hence, we fnd λ and w 1,..., w C rb wth x/ x = λ w. Snce C s a cone, we obtan ( γ C (x) = γ C x x ) ( ) x = x p γ C x p λ w p r p x p. x x Notng that r p s a constant ndependent of x, we obtan the closedness of cl seq w C from Lemmas 3.12 and 3.13 snce condton (10) s vald whch mples (4). For the proof of the converse ncluson, we assume that cl seq w C s closed. Then, by means of Lemmas 3.12 and 3.13, condton (4) holds wth some c > 0. Now, let x conv C wth x 1 be gven. Snce γ C (x) 2 c, Lemma 3.1 mples the exstence of λ, v C wth x = λ v and γ C (x) = λ v p. Now, we defne K := λ v and α := K/ v 0, = 1,..., q + 1. Ths mples x = λ α (α v ), λ λ v = α K = 1, α v = K = 1,..., q + 1. Snce C s a cone, we have α v C for all = 1,..., q + 1. Thus, x conv(c K B) s vald. It remans to fnd an upper bound for K whch s ndependent of x. Usng Hölders nequalty, we fnd wth 1 = 1/p + 1/p : K = λ v = (λ 1/p v ) λ (p 1)/p ( Usng p = p/(p 1) and λ, ths mples K ( (λ 1/p ) 1/p ( v ) p ) 1/p λ v p γ C (x) 1/p = (2 c) 1/p. Hence, (11) s satsfed wth r = (2 c) 1/p. Usng the above lemma, we obtan the followng result. (λ (p 1)/p ) p ) 1/p Corollary Suppose that (, Σ, m) satsfes Assumpton 3.5. Let C R q be a nonempty, closed cone such that conv C s a closed cone and one of the followng condtons s satsfed: () conv C s ponted (.e. we have conv C ( conv C) = {0), () conv C s polyhedral (.e. t concdes wth the conc convex hull of fntely many vectors), () q

21 We consder the decomposable set C defned n (9). Then cl seq w C s closed. Proof. For the proof, we only need to show that condton (11) holds, cf. Lemma Case (): Due to the pontedness of conv C, the polar cone has an nteror pont,.e., there s a nt C. Ths mples Clearly, we have c conv C \ {0: a c > 0. (conv C) B {x R q a x a and the set C a := {c C a c a s bounded. Consequently, there s some scalar r > 0 whch satsfes C a C rb. Now, choose x (conv C) B dfferent from 0 arbtrarly. Then, we fnd λ and v 1,..., v C \ {0 such that x = λ v. For any = 1,..., q + 1, we set β := (a x)/(a v ) > 0. Then, we have x = λ β (β v ), λ λ (a v ) = β a x = a ( λ ) v = 1, a x and β v C for all = 1,..., q + 1. Furthermore, a (β v ) = β (a v ) = a x a,.e. β v C a holds for all = 1,..., q + 1. Ths shows x conv C a conv(c rb) by constructon and, thus, (11) holds. Case (): By assumpton, there are a number N N and a set {v 1,..., v N conv C wth conv C = cone conv{v 1,..., v N. We defne Further, we set I := { I {1,..., N the famly {v I s lnearly ndependent. t := mn I I mn λ I λ v. Theren, we used I := {λ N λ = 0 for all I. Due to the lnear ndependence of the famly {v I for I I, we have t > 0. Further, we choose s > 0 such that I v conv(c s B) = 1,..., N. Now, let x (conv C) B be arbtrary. By Carathéodory s theorem, there s an ndex set I I, λ I and α 0 wth x = α I λ v. Hence, 1 x = α λ v α t. Ths shows α t 1 and from x = I λ (α v ), we fnd I x conv(α conv(c s B)) = conv(c α s B) conv(c t 1 s B). Thus, (11) s verfed wth r = t 1 s. Case (): If conv C s ponted, we can apply case (). Otherwse, let L conv C be the largest subspace contaned n conv C and set K := L (conv C). Then we obtan conv C = L + K. Snce the cone K s at most two-dmensonal, t s polyhedral. Ths shows that conv C s polyhedral, and we can apply case (). 19

22 3.4 Counterexamples In ths secton, we provde some counterexamples n whch the weak sequental closure s not closed. We gve a bref overvew of the dfferent features of these counterexamples. Example 3.16: We use the smple, two-dmensonal set C := ( R + 0 {0) {(0, 1), whch possesses a non-closed convex hull, and = (0, 1) equpped wth Lebesgue s measure. The weak sequental closure of the assocated decomposable set s strctly smaller than the decomposable set assocated wth the pontwse convex hull. Hence, ths example shows that both nclusons n (8) may be strct at the same tme. Example 3.17: We use C := {(x 1, x 2 ) R 2 x 2 x 1. Here, the convex hull conv C = R 2 s closed. Nevertheless, condton (4) s volated. Example 3.18: We use C := { 2 2n n N. Agan, the convex hull conv C = R s closed, but (4) s volated. Hence, even n R, the closedness of conv C does not mply (4). Example 3.19: We consder C = {0, 1 R whch has a closed convex hull. However, (10) s not satsfed whle (4) holds. Example 3.20: We use C := cone ( {(1, 0, 0, 0), ( 1, 0, 0, 0) {( ) 1, 1, cos(t), sn(t) t [0, 2π)). 2π t It s shown that C s a closed cone and conv C s closed, too. However, condton (11) s volated. Hence, even for closed cones, the closedness of the convex hull does not mply (4). Due to Corollary 3.15 (), we cannot construct such a counterexample n less than four dmensons. Example In ths example, we demonstrate that the nclusons n (8) mght be strct at the same tme. We consder = (0, 1) equpped wth the Lebesgue measure, p (1, ), and C := ( R + 0 {0) {(0, 1) R 2. Obvously, the assocated set C L p (m; R 2 ) defned n (9) s nonempty, closed, and decomposable. It can be checked that conv C = ( {0 [0, 1] ) ( R + [0, 1) ) holds. Furthermore, we denote by D := {v L p (m; R 2 ) v(ω) conv C f.a.a. ω the pontwse convex hull assocated to C. 20

23 Observe that we have (x 1, x 2 ) R 2 : γ C (x 1, x 2 ) = { x p 1 (1 x 2) 1 p + x 2 f (x 1, x 2 ) conv C, + f (x 1, x 2 ) conv C. For some α > 0, we consder the functon v D defned by v(ω) = (1, 1 ω α ) for ω. We obtan Γ C ( v) = = 1 0 [ (ω α ) 1 p + 1 ω α] dω 1 α (1 p) α α (1 p) + 1 lm (1 p)+1 ωα ω 0 whch dverges for α > 1/(p 1). Thus, n ths case, we obtan v cl seq w C from Theorem 3.8. However, we have cl w C = {v L p (m; R 2 ) v(ω) conv C f.a.a. ω = {v L p (m; R 2 ) v(ω) R + 0 [0, 1] f.a.a. ω, see Lemma 3.6,.e. cl seq w C D cl w C. Example We gve an example smlar to Example 3.16, but n whch the pontwse convex hull s already closed. We set C := {(x 1, x 2 ) R 2 x 2 x 1. Obvously, we have conv C = R 2. For gven p (1, ), one can compute x 2 > 0: γ C (0, x 2 ) = (x x 4 2) p 2. In order to do so, one can proceed n three steps: Frst, we show that for (λ, v) 3 C 3 wth parwse dsjont v satsfyng (0, x 2 ) = 3 λ v and λ 1, λ 2, λ 3 > 0, we have γ C (0, x 2 ) < 3 λ v p. Clearly, f v 1 belongs to the left half space whle v 2, v 3 belong to the rght half space, then there s a pont ṽ on the lne segment between v 2 and v 3 such that (0, x 2 ) belongs to the convex hull of v 1 and ṽ. Obvously, the correspondng convex combnaton yelds a better functon value n (1), snce v v p s strctly convex. Thus, we can assume that ( λ, v) 2 C 2 s a mnmzer of (1). Next, we show v 1, v 2 bd C = {(x 1, x 2 ) R 2 x 2 = x 1. Assumng v 1 / bd C on the contrary, for suffcently small ε > 0, we have ṽ := v 1 +ε( v 2 v 1 ) C. From (0, x 2 ) = λ 1 v 1 +(1 λ 1 ) v 2, we obtan (0, x 2 ) = λ 1 1 εṽ + ( 1 λ 1 1 ε) v2. Usng the strct convexty of v v p, we obtan γ C (0, x 2 ) λ 1 1 ε ṽ p + ( 1 λ 1 1 ε whch contradcts the optmalty of ( λ, v). ) v 2 p < λ 1 v 1 p + (1 λ 1 ) v 2 p = λ 1 v 1 p + λ 2 v 2 p 21

24 Fnally, we show that ( λ, v), wth λ 1 = λ 2 = 1 2, v 1 = (x 2 2, x 2), and v 2 = ( x 2 2, x 2), s actually a mnmzer of (1). Therefore, assume that a mnmzer (ˆλ, ˆv) 2 bd C 2 of (1) satsfes ˆv 1 = ( x 2 2, x 2) and ˆv 2 = ( x 2 2, x 2) for x 2, x 2 0 and x 2 x 2. Set ˆv 3 := (x 2 2, x 2), ˆv 4 := ( x 2 2, x 2) as well as µ 1 = µ 4 := ˆλ 1 2 and µ 2 = µ 3 := ˆλ 2 2. Now, we apply the frst of our above arguments to the pars ˆv 1, ˆv 3 and ˆv 2, ˆv 4 n order to fnd a feasble pont of (1) wth a better objectve value than (ˆλ, ˆv). Ths s a contradcton. Thus, a mnmzer of (1) needs to satsfy x 2 = x 2. Ths shows that ( λ, v) as presented above s a mnmzer to (1). We fx an arbtrary bounded doman R d and equp t wth Lebesgue s measure. Next, we pck a functon w { v L p (m) \ L 2p (m) v(ω) > 0 f.a.a. ω and consder v := (0, w) L p (m; R 2 ). From conv C = R 2 we fnd cl w C = L p (m; R 2 ), where C denotes the nonempty, closed, decomposable set assocated to the set C, see (9). Hence, v cl w C. However, we easly see Γ C ( v) = γ C (0, w(ω)) dω = (w 2 (ω) + w 4 (ω)) p/2 dω w 2p L 2p (m) = +. Thus, v does not belong to cl seq w C. Example Let us consder the set C := { 2 2n n N, p (1, ), and the assocated pontwse defned set C L p (m) gven n (9). Here, R d s an arbtrary bounded doman whch s equpped wth Lebesgue s measure. For arbtrary n N, we defne α n := 2 (p+1) 2n 1 (0, 1) and Then, we can check w n := (1 α n ) 2 2n + α n 2 2n+1. γ C (w n ) = (1 α n ) 2 p 2n + α n 2 p 2n+1 α n 2 p 2n+1 = 2 (3 p 1) 2n 1 = 2 (p 1) 2n 1 2 p 2n as well as w n p 2 p 1 [ (1 α n ) p 2 p 2n + α p n 2 p 2n+1] 2 p 1 2 p 2n [ 1 + α p n 2 p 2n ] 2 p 2 p 2n, and takng the lmt n, we obtan 2n 1 γ C (w n ) 2(p 1) w n p 2 p = 2 (p 1) 2n 1 p +. 22

25 Together wth the trval estmate we even get w n (p 1)/8 2 (p 1) 2n 2, γ C (w n ) w n p+(p 1)/8 2(p 1) 2n 1 p (p 1) 2 n 2 = 2 (p 1) 2n 2 p 1, where the last nequalty holds for n N where N N s chosen large enough. Now, choose a functon { v v L p (m) \ L p+(p 1)/8 (m) v(ω) {w n n N, n N f.a.a. ω. Snce we have conv C = [2, ), v s an element of cl w C. On the other hand, Γ C ( v) = γ C ( v(ω)) dω v(ω) p+(p 1)/8 dω = v p+(p 1)/8 L p+(p 1)/8 (m) = + s obtaned. Thus, v does not belong to cl seq w Theorem 3.9. C whch, therefore, cannot be closed by Example Let us consder the unbounded doman = (1, ) equpped wth Lebesgue s measure, p (1, ), and the closed set C := {0, 1 R. Clearly, we have conv C = conv C = [0, 1] and γ C (x) = x for all x conv C. Consequently, the condton (4) s vald whle (10) s volated,.e. cl seq w C, where C L p (m) denotes the assocated decomposable set defned n (9), s not closed. Let us vsualze ths by means of an example. We have cl w C = {v L p (m) v(ω) [0, 1] f.a.a. ω from Lemma 3.6. Let us consder the functon v : R defned by v(ω) := ω 1 for any ω. Due to v p L p (m) = ω p dω = 1 ( ) 1 lm 1 p ω ω p 1 1 = 1 p 1, we have v cl w C. On the other hand, Γ C ( v) = 1 1 γ C ( v(ω)) dω = 1 ω 1 dω = lm ln ω = +. ω Followng Theorem 3.8, we have v / cl seq w C and, thus, cl seq w C s not closed by Theorem 3.9. Example We defne ( {( ) {(1, 1 C := cone 0, 0, 0), ( 1, 0, 0, 0), 1, cos(t), sn(t) t [0, 2 π)) R 4. 2π t 23

26 Let us check that C s closed. It s suffcent to consder convergent sequences {x n n N of the form ( ) 1 x n = α n, 1, cos(t n ), sn(t n ) 2π t n wth {α n n N (0, ) and {t n n N [0, 2 π). By consderng the second component of {x n, we obtan the boundedness of α n, thus, α n α holds along a subsequence. Snce {t n s bounded, we fnd t n t along a subsequence. If α = 0, we fnd lm x n = lm n n ( α n, 0, 0, 0 (2π t n ) On the other hand, α > 0 mples t 2 π. Ths yelds lm x n = α n ( 1, 1, cos(t), sn(t) 2π t Ths shows the closedness of C. Next, we compute the convex hull of C. We have ) C. ) C. conv C = conv cone ({ (1, 0, 0, 0), ( 1, 0, 0, 0) {(0, 1, cos(t), sn(t)) t [0, 2 π) ) = ( R {0 3) + ( {0 L ), where L := { (x 1, x 2, x 3 ) R 3 x1 x x2 3 s the Lorentz cone n R 3. Hence, conv C s closed. Fnally, we consder the ponts y n = (0, 1, cos(t n ), sn(t n )) wth t n = 2π 1/n. It s clear that y n conv(c) 2 B s vald. Now, we are wrtng y n as a convex combnaton of vectors from C,.e. M n y n = λ n c n, wth λ n > 0, c n C, and M n λn = 1 for all = 1,..., M n and n N. It s clear that none of the c n can be of the form ( ) 1 α, 1, cos(t), sn(t) 2π t wth t t n. Thus, we can wrte c n = α n (1, 0, 0, 0) for = 1,..., M 1 n, c n = α n ( 1, 0, 0, 0) for = M 1 n + 1,..., M 2 n, c n = α n (n, 1, cos(t n ), sn(t n )) for = M 2 n + 1,..., M n 24

27 for scalars α n 0, = 1,..., M n, and n N. We set α n := max{α n = 1,..., M n. By consderng the second component of y n, we get The frst component of y n yelds M 1 n 0 = λ n α n Thus, we obtan M 2 n =M 1 n+1 λ n α n + 1 = α n M n =M 2 n +1 λ n α n. M n =M 2 n+1 M 2 n =M 1 n+1 M 1 n λ n α n n = λ n α n λ n α n n. M 2 n =M 1 n+1 λ n α n + n. Hence, y n conv(c r B) s vald for r < n. Takng the lmt n shows that (11) s volated. By means of Lemma 3.14, the weak sequental closure of the assocated decomposable set C L p (m; R 4 ) defned n (9) cannot be closed. 4 Weak sequental closure n purely atomc measure spaces If (, Σ, m) s σ-addtve and purely atomc, there are at most countably many nonequvalent atoms (w.r.t. the equvalence relaton defned by A B m(a B) = 0). In the case that the measure space contans only fntely many non-equvalent atoms, the spaces L p (m) wll be fnte dmensonal and, thus, are sometrc to some R n wth a weghted p-norm. Hence, we wll assume that there are countably many atoms {A k k N Σ. Assumpton 4.1. We assume that (, Σ, m) s a σ-fnte, complete, and purely atomc measure space, and we fx p (1, ). Moreover, we assume that there are countably many non-equvalent atoms {A k k N Σ. Recall that a measurable functon u: R s constant a.e. on every atom A k. We denote by u Ak R the value of u on the atom A k. Ths mples that L p (m) s sometrc to the sequence space L p (n), where the measure n: 2 N R + 0 s gven by n(n) = k N m(a k) for all N N. The sometrc somorphsm I : L p (m) L p (n) s gven by (I(u))(k) = u Ak. Next, we wll demonstrate the well-known fact that weak convergence mples pontwse convergence on purely atomc spaces. Lemma 4.2. Assume that Assumpton 4.1 s satsfed. Let {u I L p (m) be a gven net over the drected set I such that u u L p (m). Then, u u a.e. on. 25

28 Proof. Due to m(a k ) < +, we have χ Ak L p (m). Hence, we obtan u Ak m(a k ) = u χ Ak dω u χ Ak dω = u Ak m(a k ) for all k N. Ths shows the desred pontwse convergence. As a corollary, we fnd that the weak closure s qute well-behaved for decomposable sets n Lebesgue spaces over a purely atomc measure. Corollary 4.3. Suppose that Assumpton 4.1 s satsfed and that C L p (m; R q ) s nonempty and decomposable. Then, C s closed f and only f C s weakly closed. Moreover, we obtan cl w C = cl seq w C = cl C. Proof. In order to verfy the frst asserton, we have to show that the closedness of C mples that C s weakly closed. Snce C s closed, we know that t possesses the representaton C = {u L p (m; R q ) u(ω) C(ω) f.a.a. ω, where the assocated measurable set-valued map C : R q has nonempty and closed mages. If the net {u I converges weakly to u n L p (m; R q ), we can nvoke Lemma 4.2 and obtan u(ω) = lm I u (ω) C(ω) by closedness of C(ω). Hence, u C holds true and, thus, C s weakly closed. Let us prove the second asserton. We obvously have cl C cl seq w C cl w C. By applyng the weak closure to C cl C, we fnd cl w C cl w cl C = cl C snce cl C s closed and decomposable, thus weakly closed by the frst asserton. Ths proves the equaltes cl C = cl seq w C = cl w C. Agan, we emphasze that ths result s qute surprsng snce the weak topology on nfnte-dmensonal Banach spaces does not cooperate well wth sequences. However, Corollary 4.3 demonstrates that ths does not happen for decomposable sets. Fnally, we note that Corollary 4.3 s also true f the measure space contans only fntely many atoms, snce L p (m) s fnte dmensonal n ths case. Hence, t s true for all σ-fnte, complete, and purely atomc measure spaces. 5 Varatonal objects assocated wth decomposable sets Throughout ths secton, we postulate the followng standng assumpton. Assumpton 5.1. We assume that (, Σ, m) s a σ-fnte, complete, and separable measure space. Furthermore, let K L p (m; R q ) be a nonempty, closed, and decomposable set wth p (1, ). By K : R q, we denote the assocated measurable and closed-valued multfuncton. We assume that the mages of K are dervable almost everywhere n. Let p (1, ) be the conjugate coeffcent assocated to p,.e. the scalar whch satsfes 1/p + 1/p = 1. Fnally, we fx some pont ū K. 26