THE MCSHANE INTEGRAL IN WEAKLY COMPACTLY GENERATED SPACES. 1. Introduction

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1 THE MCSHANE INTEGRAL IN WEAKLY COMPACTLY GENERATED SPACES A. AVILÉS, G. PLEBANEK, AND J. RODRÍGUEZ Abstract. D Pazza and Press asked whether every Petts ntegrable functon defned on [0, 1] and takng values n a weakly compactly generated Banach space s McShane ntegrable. In ths paper we answer ths queston n the negatve. 1. Introducton The classcal Petts measurablty theorem [15] ensures that scalar and strong measurablty are equvalent for functons takng values n separable Banach spaces. Ths fact has many nterestng consequences n vector ntegraton. For nstance, t s a basc tool to prove that Petts and McShane ntegrablty concde n separable Banach spaces, [10, 12, 13]. However, for non-separable Banach spaces the notons of scalar and strong measurablty are dfferent n general. Ths leads to subtle problems when tryng to compare dfferent types of ntegrals. In ths paper we deal wth the Petts and McShane ntegrals. D Pazza and Press [2] asked whether every Petts ntegrable functon f : [0, 1] X s McShane ntegrable f X s a weakly compactly generated (WCG Banach space. Recently, Devlle and the thrd author [1] have proved that the answer s affrmatve when X s Hlbert generated, thus mprovng the prevous results obtaned n [2, 16]. Our man purpose here s to show that the queston of D Pazza and Press has negatve answer n general. The paper s organzed as follows. In Secton 2 we ntroduce the MC-ntegral for Banach space-valued functons defned on probablty spaces. Ths auxlary tool s used as a substtute of the McShane ntegral at some stages. We prove that, for functons defned on quas- Radon probablty spaces, MC-ntegrablty always mples McShane ntegrablty (Proposton 2.2, whle the converse holds f the topology on the doman has a countable bass (Proposton 2.3. Ths approach allows us to gve a partal answer (Corollary 2.4 to a queston posed by Fremln n [10, 4G(a] Mathematcs Subject Classfcaton. 28B05, 46G10. Key words and phrases. Petts ntegral; McShane ntegral; scalarly null functon; fllng famly. A. Avlés and J. Rodríguez were supported by MEC and FEDER (Project MTM and Fundacón Séneca (Project 08848/PI/08. A. Avlés was supported by Ramon y Cajal contract (RYC G. Plebanek wshes to thank A. Avlés, B. Cascales and J. Rodríguez for ther hosptalty durng hs stay n Murca n February 2009; the vst was supported by Departamento de Matemátcas, Unversdad de Murca. 1

2 2 A. AVILÉS, G. PLEBANEK, AND J. RODRÍGUEZ In Secton 3 we show that the exstence of scalarly null (hence Petts ntegrable WCG-valued functons whch are not McShane ntegrable s strongly related to the exstence of famles of fnte sets whch are measure fllng n the sense of the followng defnton. Throughout the paper (Ω, Σ, µ s a probablty space and we use the symbol [S] <ω to denote the famly of all fnte subsets of a gven set S. Defnton 1.1. A heredtary famly F [Ω] <ω s MC-fllng on Ω f there exsts ε > 0 such that for every countable ( partton (Ω m of Ω there s F F such that µ {Ωm : F Ω m } > ε, where µ s the outer measure nduced by µ. Ths concept should be vewed as a measure-theoretc analogue of the noton of ε-fllng famles arsng n Fremln s problem DU [8]: Defnton 1.2. Let ε > 0. A heredtary famly F [S] <ω s ε-fllng on the set S f for every H [S] <ω there s F F wth F H and F ε H. The exstence of compact ε-fllng famles on uncountable sets s an open problem (the above mentoned problem DU. However, we show that compact MC-fllng famles on [0, 1] can be constructed from some weaker versons of fllng famles that Fremln proved to exst (Theorem 3.4. Ths leads to our man result: Theorem 3.5. There exst a WCG Banach space X and a scalarly null functon f : [0, 1] X whch s not McShane ntegrable. In fact, the space X can be taken reflexve (Theorem 3.6. Observe that Theorem 3.5 also answers n the negatve the queston (attrbuted to Musal n [2] whether every scalarly null Banach space-valued functon on [0, 1] s McShane ntegrable. Two counterexamples [2, 16] had been constructed under the Contnuum Hypothess (and havng non WCG spaces n the range. In Secton 4 we prove that f a famly F [A] <ω s ε-fllng on a set A Ω of postve outer measure then t s MC-fllng on Ω. Fnally, n Secton 5 we provde an example of a McShane ntegrable functon whch s not MC-ntegrable (Theorem 5.5. Our example also makes clear that, n general, the results on the concdence of Petts and McShane ntegrablty of [1, 2] do not hold when McShane ntegrablty s replaced by MC-ntegrablty. Termnology. Our standard references are [4, 18] (vector ntegraton and [11] (topologcal measure theory. By a partton of a set S we mean a collecton of parwse dsjont (maybe empty subsets whose unon s S. A set s countable f t s ether fnte or countably nfnte. As usual, the symbol S stands for the cardnalty of a set S. A famly F [S] <ω s heredtary f G F whenever G F F. A famly F [S] <ω s called compact f t s compact n 2 S equpped wth the product topology. We say that a famly E Σ s η-thck (for some η > 0 f µ(ω \ E η. Throughout the paper X s a (real Banach space. The norm of X s denoted by f t s needed explctly. We denote by X the topologcal dual of X and B X = {x X : x 1}. The space X s WCG f there s a weakly compact

3 THE MCSHANE INTEGRAL IN WEAKLY COMPACTLY GENERATED SPACES 3 subset of X whose lnear span s dense n X. Recall that a functon f : Ω X s called scalarly null f, for each x X, the composton x f : Ω R vanshes µ-a.e. (the exceptonal set dependng on x. If T Σ s a topology on Ω, we say that (Ω, T, Σ, µ s a quas-radon probablty space (followng [11, Chapter 41] f µ s complete, nner regular wth respect to closed sets, and µ( G = sup{µ(g : G G} for every upwards drected famly G T. A gauge on Ω s a functon δ : Ω T such that t δ(t for all t Ω. Every Radon probablty space s quas-radon [11, 416A]. The vector-valued McShane ntegral was frst studed n [12, 13] for functons defned on [0, 1] equpped wth the Lebesgue measure. Fremln [10] extended the theory to deal wth functons defned on arbtrary quas-radon probablty spaces. We next recall an alternatve way of defnng the McShane ntegral taken from [9, Proposton 3]. Defnton 1.3. Suppose (Ω, T, Σ, µ s quas-radon. A functon f : Ω X s McShane ntegrable, wth ntegral x X, f for every ε > 0 there exst η > 0 and a gauge δ on Ω such that: for every η-thck fnte famly (E of parwse dsjont measurable sets and every choce of ponts t Ω wth E δ(t, we have µ(e f(t x ε. Every McShane ntegrable functon s also Petts ntegrable (and the correspondng ntegrals concde, [10, 1Q]. The converse does not hold n general, see [1, 2, 12, 16] for examples. 2. Another look at the McShane ntegral We next ntroduce a varant of the McShane ntegral that s defned n terms of the measure space only, wthout any reference to a topology. Defnton 2.1. A functon f : Ω X s MC-ntegrable, wth ntegral x X, f for every ε > 0 there exst η > 0, a countable partton (Ω m of Ω and sets A m Σ wth Ω m A m, such that: for every η-thck fnte famly (E of parwse dsjont elements of Σ wth E A m( and every choce of ponts t Ω m(, we have µ(e f(t x ε. Clearly, gven η > 0, a countable partton (Ω m of Ω and sets Ω m A m Σ, we can always fnd famles (E as n Defnton 2.1. It s routne to check that the vector x n Defnton 2.1 s unque. The relatonshp between the MC-ntegral and the McShane ntegral s analyzed n the followng two propostons. Proposton 2.2. Suppose (Ω, T, Σ, µ s quas-radon. If f : Ω X s MCntegrable, then t s McShane ntegrable (and the correspondng ntegrals concde.

4 4 A. AVILÉS, G. PLEBANEK, AND J. RODRÍGUEZ Proof. Let x X be the MC-ntegral of f and fx ε > 0. Snce f s MC-ntegrable, there exst η > 0, a countable partton (Ω m of Ω and measurable sets A m Ω m satsfyng the condton of Defnton 2.1. For each m, n N, set Ω m,n := {t Ω m : n 1 f(t < n} and choose U m,n A m open such that µ(u m,n \ A m 1 { ε 2 m+n mn n, η }. 2 Clearly, (Ω m,n s a partton of Ω. Defne a gauge δ : Ω T by δ(t := U m,n f t Ω m,n. Let (E be a η 2 -thck fnte famly of parwse dsjont measurable sets and let t Ω be ponts such that E δ(t. We wll check that (1 µ(e f(t x 2ε. For each, let m(, n( N be such that t Ω m(,n(. The set F := E A m( s measurable, F A m( and t Ω m(. The F s are parwse dsjont. Snce we have ( µ Ω \ E \ F = E \ A m( δ(t \ A m( = U m(,n( \ A m( ( F = µ Ω \ ( E + µ E \ F η ( 2 + µ U m,n \ A m η 2 + η = η, 2m+n+1 m,n m,n and so the famly (F s η-thck. From the MC-ntegrablty condton t follows that (2 µ(f f(t x ε. For each m, n N, let I(m, n be the (maybe empty set of all ndexes for whch m( = m and n( = n. Observe that µ(e \ F f(t µ(e \ F n = I(m,n Therefore (3 µ(e f(t I(m,n ( = µ I(m,n Inequalty (1 now follows from (2 and (3. ntegrable, wth McShane ntegral x. E \ F n µ(u m,n \ A m n ε 2 m+n. µ(f f(t µ(e \ F f(t = = µ(e \ F f(t ε = ε. 2m+n m,n I(m,n m,n Ths shows that f s McShane

5 THE MCSHANE INTEGRAL IN WEAKLY COMPACTLY GENERATED SPACES 5 The converse of Proposton 2.2 does not hold n general (see Theorem 5.5 below, although t s true for certan quas-radon spaces lke [0, 1], as we next prove. Proposton 2.3. Suppose (Ω, T, Σ, µ s quas-radon and T has a countable bass. Then f : Ω X s McShane ntegrable f and only f t s MC-ntegrable. Proof. It only remans to prove the only f. Assume that f s McShane ntgerable, wth McShane ntegral x X. Let {U m : m N} be a countable bass for T. Fx ε > 0. Snce f s McShane ntegrable, there exst η > 0 and a gauge δ on Ω fulfllng the condton of Defnton 1.3. We can suppose wthout loss of generalty that δ(t {U m : m N} for every t Ω. Set Ω m := {t Ω : δ(t = U m } and A m := U m for all m N. Clearly, (Ω m s a partton of Ω and Ω m A m Σ. Now let (E be an η- thck fnte famly of parwse dsjont measurable sets wth E A m( and let t Ω m(. Then δ(t = U m( = A m(, hence E δ(t for all. From the McShane ntegrablty condton t follows that µ(e f(t x ε. Ths shows that f s MC-ntegrable. Fremln rased n [10, 4G(a] the followng queston: Does any topology on Ω for whch µ s quas-radon yeld the same collecton of McShane ntegrable X-valued functons? In vew of Propostons 2.2 and 2.3, we get a partal answer: Corollary 2.4. Let T 1 and T 2 be two topologes on Ω for whch µ s quas-radon. Suppose T 1 has a countable bass. If f : Ω X s McShane ntegrable wth respect to T 1, then t s also McShane ntegrable wth respect to T 2 (and the correspondng ntegrals concde. 3. MC-fllng famles vs the McShane ntegral The connecton between MC-fllng famles (Defnton 1.1 and the MC-ntegral s explaned n Proposton 3.2 below. Frst, t s convenent to characterze MCfllng famles as follows: Lemma 3.1. A famly F [Ω] <ω s MC-fllng on Ω f and only f there exsts ε > 0 such that for every countable partton (Ω m of Ω and sets A m Σ wth Ω m A m, there s F F such that ( µ {Am : F Ω m } > ε. Proof. The only f s obvous. For the converse, we wll prove that the condton of Defnton 1.1 holds for 0 < η < ε. Suppose we are gven a countable partton (Ω m of Ω. For every fnte set I N, we choose B I Σ such that B I m I Ω m and µ(b I µ ( m I Ω m < ε η. For each m N, we defne A m := {B I : m I}. We have Ω m A m Σ, so we can apply the hypothess to fnd F F such that ( µ {Am : F Ω m } > ε.

6 6 A. AVILÉS, G. PLEBANEK, AND J. RODRÍGUEZ Consder the fnte set I := {m N : F Ω m }. Snce m I A m B I, we have µ ( ( Ω m > µ(b I (ε η µ A m (ε η > η. m I Ths proves that F s MC-fllng. A set B B X s called normng f x = sup{ x (x : x B} for all x X. As usual, gven a set C Ω, we wrte 1 C to denote the real-valued functon on Ω defned by 1 C (t = 1 f t C and 1 C (t = 0 f t C. Proposton 3.2. Let f : Ω X be a functon for whch there exst a normng set B B X and a famly (C x x B of subsets of Ω such that x f = 1 Cx and µ (C x = 0 for every x B. The followng statements are equvalent: ( f s not MC-ntegrable; ( x B [C x ]<ω s MC-fllng on Ω. Proof. Observe frst that for every fnte famly (E of parwse dsjont elements of Σ and every choce of ponts t Ω, we have (4 µ(e f(t = sup x ( µ(e f(t = x B = sup x B m I ( µ(e 1 Cx (t = sup µ {E : t C x }. x B Snce B separates the ponts of X and x f vanshes µ-a.e. for each x B, the MC-ntegral of f s 0 X whenever f s MC-ntegrable. Bearng n mnd (4, statement ( s equvalent to: ( There exsts ε > 0 such that for every η > 0, every countable partton (Ω m of Ω and sets A m Σ wth Ω m A m, there exst an η-thck fnte famly (E of parwse dsjont elements of Σ wth E A m(, ponts t Ω m( and a functonal x B such that µ ( {E : t C x } > ε. Let us turn to the proof of ( (. Assume frst that ( holds and take a countable partton (Ω m of Ω and sets Ω m A m Σ. Choose η > 0 arbtrary and let (E, (t and x be as n (. Observe that the set F made up of all t s belongng to C x satsfes {Am : F Ω m } {E : t C x } and so µ( {A m : F Ω m } > ε. Accordng to Lemma 3.1, ths proves that the famly x B [C x ]<ω s MC-fllng on Ω. Conversely, assume that ( holds. Let ε > 0 be as n Lemma 3.1 appled to the famly x B [C x ]<ω. Fx η > 0, a countable partton (Ω m of Ω and sets A m Σ wth Ω m A m. There exst x B and F C x fnte such that µ( m I A m > ε, where I := {m N : F Ω m }. Now take a fnte set J N dsjont from I such that (A m m I J s η-thck. Enumerate I = {m(1,..., m(n} and J = {m(n + 1,..., m(k}. Set E 1 := A m(1 and E := A m( \ 1 j=1 A m(j for = 2,..., k. Then (E s an η-thck fnte famly of parwse dsjont elements of Σ

7 THE MCSHANE INTEGRAL IN WEAKLY COMPACTLY GENERATED SPACES 7 wth E A m( and n =1 E = m I A m. Choose t F Ω m( for = 1,..., n and choose t Ω m( arbtrary for = n + 1,..., k. Then ( µ( n {E : t C x } µ =1 ( E = µ m I Ths shows that ( holds, that s, f s not MC-ntegrable. A m > ε. Gven a compact Hausdorff topologcal space K, we wrte C(K to denote the Banach space of all real-valued contnuous functons on K (wth the sup norm. Proposton 3.3. Let F [Ω] <ω be a compact heredtary famly made up of sets havng outer measure 0. Let f : Ω C(F be defned by f(t(f := 1 F (t. Then: ( f s scalarly null; ( f s not MC-ntegrable f and only f F s MC-fllng on Ω. Proof. Part ( follows from a standard argument whch we nclude for the sake of completeness. Snce F s an Eberlen compact (.e., t s homeomorphc to a weakly compact subset of a Banach space, the space C(F s WCG and B C(F s Eberlen compact when equpped wth the w -topology, cf. [3, Theorem 4, p. 152]. Set B := {δ F : F F} B C(F, where δ F denotes the evaluaton functonal at F. Snce B s normng, ts absolutely convex hull aco(b s w -dense n B C(F. Bearng n mnd that (B C(F, w s homeomorphc to a weakly compact subset of a Banach space, the Eberlen-Smulyan theorem (cf. [7, 3.10] ensures that aco(b s w -sequentally dense n B C(F. Snce the composton δ F f = 1 F vanshes µ- a.e. for every F F, we conclude that f s scalarly null. Part ( follows from Proposton 3.2 appled to f and the normng set B defned above. On the other hand, t turns out that we can always fnd compact MC-fllng famles on [0, 1]. As usual, we denote by c the cardnalty of the contnuum. Theorem 3.4. There exsts a compact MC-fllng famly on [0, 1] equpped wth the Lebesgue measure. Proof. Accordng to a result by Fremln [8, 4B], there s a famly D [c] <ω whch s heredtary and compact, and for every nonempty A [c] <ω there s D D such that D A and D log A. In partcular, D has the followng property: (* If P c s nfnte, then for every n N there s D D such that D P and D = n. We denote by λ the Lebesgue measure on [0, 1]. Fx a partton {Z α : α < c} of [0, 1] made up of sets of outer measure one (cf. [11, 419I]. Let ϕ : [0, 1] c be the functon defned by ϕ(t = α whenever t Z α. We defne the famly F := { F [0, 1] fnte : ϕ s one-to-one on F and ϕ(f D }. It s clear that F s heredtary (because D s. We clam that F s compact or, equvalently, that every set A [0, 1] wth [A] <ω F s fnte. Indeed, observe that ϕ s one-to-one on A. Gven any C [ϕ(a] <ω, we have C = ϕ(b for some

8 8 A. AVILÉS, G. PLEBANEK, AND J. RODRÍGUEZ B [A] <ω F and so C D. Hence [ϕ(a] <ω D and the compactness of D ensures that ϕ(a s fnte. Snce ϕ s one-to-one on A, we conclude that A s fnte. We shall check that F s MC-fllng on [0, 1] wth arbtrary constant 0 < ε < 1. Fx a countable partton (Ω m of [0, 1]. For each α < c we have so we can pck n(α N such that 1 = λ (Z α = lm n λ ( Z α (5 λ ( Z α n(α m=1 n m=1 Ω m > ε. Ω m, Fx n N such that P n := {α < c : n(α = n} s nfnte. By property (*, there s D D such that D P n and D = n. Wrte D = {α 1,..., α n }. We next defne t j Z αj and m j {1,..., n} nductvely as follows. By (5 the set Z α1 n m=1 Ω m s nonempty and we pck any t 1 Z α1 n m=1 Ω m. Choose m 1 {1,..., n} so that t 1 Ω m1. Now suppose we have already constructed a set {m 1,..., m l } {1,..., n} and ponts t j Z αj Ω mj for j = 1,..., l. If λ ( l j=1 Ω m j > ε, the constructon stops. Otherwse λ ( l j=1 Ω m j ε and therefore l < n (bear n mnd that λ ( n m=1 Ω m > ε by (5. Wrtng N := {1,..., n} \ {m 1,..., m l }, another appeal to (5 yelds λ ( Z αl+1 Ω m λ ( n Z αl+1 Ω m λ ( l Z αl+1 Ω mj > 0, m N m=1 so we can fnd t l+1 Z αl+1 Ω ml+1 for some m l+1 N. Repeatng the process, the constructon stops for some l {1,..., n}. After that, we obtan a set {m 1,..., m l } {1,..., n} wth λ ( l j=1 Ω m j > ε and ponts t j Z αj Ω mj for all j = 1,..., l. Puttng F := {t 1,..., t l } we have λ ( {Ωm : F Ω m } = λ ( l j=1 Ω mj > ε. Snce ϕ(t j = α j for all j, t follows that ϕ s one-to-one on F and ϕ(f D, thus ϕ(f D and so F F. The proof s complete. We now arrve at our man result: Theorem 3.5. There exst a WCG Banach space X and a scalarly null functon f : [0, 1] X whch s not McShane ntegrable. Proof. By Theorem 3.4, there s a compact MC-fllng famly F on [0, 1]. As we observed n the proof of Proposton 3.3, the space X := C(F s WCG. The functon f : [0, 1] C(F, f(t(f := 1 F (t, from Proposton 3.3 s scalarly null and fals to be MC-ntegrable. Accordng to Proposton 2.3, f s not McShane ntegrable. Moreover, the Banach space n the range can be taken reflexve: j=1

9 THE MCSHANE INTEGRAL IN WEAKLY COMPACTLY GENERATED SPACES 9 Theorem 3.6. There exst a reflexve Banach space Y and a scalarly null functon g : [0, 1] Y whch s not McShane ntegrable. Proof. Let F and f be as n the proof of Theorem 3.5. Observe frst that f([0, 1] s relatvely weakly compact n C(F. Indeed, by the Eberlen-Smulyan theorem (cf. [7, 3.10], t s enough to check that (f(t n converges weakly to 0 whenever (t n s a sequence of dstnct ponts of [0, 1]. But ths follows drectly from Grothendeck s theorem (cf. [7, 4.2] just bearng n mnd that for each F F (fnte! we have f(t n (F = 1 F (t n = 0 for n large enough. Then, by the Davs-Fgel-Johnson-Pelczynsk theorem (cf. [3, Chapter 5, 4], there exst a reflexve Banach space Y and a one-to-one lnear contnuous mappng T : Y C(F such that f([0, 1] T (B Y. The set of compostons V := {φ T : φ C(F } s a lnear subspace of Y whch separates the ponts of Y (because T s one-toone. Snce Y s reflexve, V s norm dense n Y. Let g : [0, 1] Y be the functon satsfyng T g = f. For each y V the composton y g vanshes a.e. (f s scalarly null. Ths fact and the norm densty of V mply that g s scalarly null. Moreover, snce f s not McShane ntegrable and T s lnear and contnuous, g s not McShane ntegrable ether. Remark 3.7. A glance at the proof of Proposton 3.3 reveals that the functon f from Theorem 3.5 satsfes that, for each x X, the composton x f vanshes up to a countable set. Ths property and the boundedness of f ensure that f s unversally Petts ntegrable, that s, Petts ntegrable wth respect to any Radon probablty on [0, 1]. The same holds true for the functon g from Theorem 3.6. Thus, we answer Queston 2.2 n [17]: there exst ZFC examples of unversally Petts ntegrable functons whch are not unversally McShane ntegrable. 4. Fllng vs MC-fllng famles In ths secton we prove that ε-fllng famles (Defnton 1.2 on sets of postve outer measure are MC-fllng. Ths result s less powerful than Theorem 3.4, n the sense that the exstence of ε-fllng famles on uncountable sets s unknown whle Theorem 3.4 s a ZFC result. Yet, we have decded to nclude t as t may have some nterest n relaton wth problem DU. Theorem 4.1. Suppose µ s atomless. Let A Ω wth µ (A > 0 and F [A] <ω be a famly whch s ε-fllng on A for some ε > 0. Then F s MC-fllng on Ω. Proof. Wrte η := µ (A and fx η > η 1 > η 2 > 0. We take a countable partton (Ω m of Ω and sets A m Ω m wth A m Σ. We wll prove that there s F F such that ( µ {Am : F Ω m } > ε(η η 1. Accordng to Lemma 3.1, ths means that F s MC-fllng on Ω.

10 10 A. AVILÉS, G. PLEBANEK, AND J. RODRÍGUEZ To ths end, take m 0 N large enough such that (6 µ (A µ ( A m 0 m=1 Ω m < η 2. Snce µ s atomless, every fnte subset of Ω has outer measure 0, so we can assume wthout loss of generalty that A Ω m s nfnte for all m = 1,..., m 0. Take 0 < η 3 < (η 1 η 2 /m 0. We can fnd parwse dsjont B 1,..., B m0 Σ such that m 0 m=1 B m = m 0 m=1 A m and B m A m. Let M be the set of all m {1,..., m 0 } for whch µ(b m > 0. For each m M, choose a postve ratonal α m such that µ(b m > α m > µ(b m η 3. We can wrte α m = p m /q for some p m N and q N, for m = 1,..., m 0. Set θ := 1/q. Snce µ s atomless, for each m M we can fnd parwse dsjont E1 m,..., Ep m m Σ contaned n B m wth µ(e m = θ. Then and we have µ ( A m 0 m=1 Ω m µ ( A ( M η 3 + ( p m µ B m \ m 0 ( µ B m \ m M m M =1 m=1 p m E m =1 p m θ m 0 η 3 + E m < η 3 A m = µ ( A + m M ( m M m M p m =1 From these nequaltes and (6 we obtan ( (7 η = µ (A < η 1 + p m θ. B m µ(e m ( p m θ < (η 1 η 2 + p m θ. m M m M For each m M and = 1,..., p m we pck a pont t (m, A Ω m. Ths can be done n such a way that the ponts t (m, s are dfferent, snce A Ω m s nfnte for all m M. Now H := {t (m, : m M, = 1,..., p m } s a subset of A wth cardnalty m M p m. Snce F s ε-fllng on A, there exsts F H wth F F such that F ε( m M p m. By (7, we get µ( {Am : F Ω m } µ( {E m : t (m, F } ( = F θ ε m M p m θ > ε(η η 1. The proof s over.

11 THE MCSHANE INTEGRAL IN WEAKLY COMPACTLY GENERATED SPACES McShane ntegrablty vs MC-ntegrablty Ths secton s devoted to ensure the exstence of McShane ntegrable functons whch are not MC-ntegrable (Theorem 5.5. The proof s dvded nto several auxlary lemmas. The frst one translates the problem nto the language of MCfllng famles. Lemma 5.1. Let Γ be a set. The followng statements are equvalent: ( there exsts a scalarly null functon f : Ω c 0 (Γ whch s not MCntegrable and satsfes f(ω {e γ : γ Γ}, where e γ (γ = δ γ,γ (the Kronecker symbol for all γ, γ Γ; ( there exsts a partton (C γ γ Γ of Ω nto sets havng outer measure 0 such that the famly γ Γ [C γ] <ω s MC-fllng on Ω. Proof. The set B := {e γ : γ Γ} B c0(γ s normng, where e γ(x = x(γ for all x c 0 (Γ and γ Γ. ( ( For each γ Γ we have e γf = 1 Cγ, where C γ := {t Ω : f(t = e γ } has outer measure 0 (because f s scalarly null. Clearly, (C γ γ Γ s a partton of Ω. Snce f s not MC-ntegrable, we can apply Proposton 3.2 to conclude that the famly γ Γ [C γ] <ω s MC-fllng on Ω. ( ( Defne f : Ω c 0 (Γ by f(t := e γ whenever t C γ, γ Γ. Then e γf = 1 Cγ for all γ Γ and f s scalarly null, because µ (C γ = 0 for all γ Γ and the lnear span of {e γ : γ Γ} s norm dense n c 0 (Γ = l 1 (Γ. By Proposton 3.2, snce γ Γ [C γ] <ω s MC-fllng on Ω, the functon f s not MC-ntegrable. Thus, bearng n mnd that Petts and McShane ntegrablty are equvalent for c 0 (Γ-valued functons [2], n order to fnd McShane ntegrable functons whch are not MC-ntegrable we wll look for MC-fllng famles lke n condton ( of Lemma 5.1. The followng suffcent condton wll be helpful. Lemma 5.2. Let (C γ γ Γ be a partton of Ω and ε > 0 such that, whenever (Γ A A N s a partton of Γ, there s some A N such that µ ( γ Γ A C γ > ε. Then the famly γ Γ [C γ] <ω s MC-fllng on Ω. Proof. Fx a countable partton (Ω m of Ω. For each A N, set Γ A := {γ Γ : C γ Ω m m A}. Then (Γ A A N s a partton of Γ and so there s A N such that µ ( γ Γ A C γ > ε. Observe that Ω m C γ, m A γ Γ A hence µ ( m A Ω m > ε. Choose B A fnte wth µ ( m B Ω m > ε. Take γ Γ A. We can fnd a fnte set F C γ such that F Ω m for every m B, hence µ ( {Ωm : F Ω m } µ ( Ω m > ε. Ths shows that γ Γ [C γ] <ω s MC-fllng on Ω. m B

12 12 A. AVILÉS, G. PLEBANEK, AND J. RODRÍGUEZ We now focus on 2 κ (for a cardnal κ, whch s a Radon probablty space when equpped wth (the completon of the usual product probablty, cf. [11, 416U]. Lemma 5.3. Let κ be an uncountable cardnal, (A α α<κ a partton of κ nto nfnte sets and consder, for each α < κ, the sets D α := {x 2 κ : x(γ = 0 for all γ A α } and E α := D α \ D β. Then α I E α has outer measure 1 for every uncountable set I κ. Proof. It suffces to check that Z ( α I E α whenever Z belongs to the product σ-algebra of 2 κ and has postve measure. Fx a countable set A κ such that, for any z Z, we have (8 {x 2 κ : x(γ = z(γ for all γ A} Z. Snce the A α s are dsjont, the set J := {α < κ : A A α } s countable. Clearly, the D α s have measure zero (because A α s nfnte and so Z \ α J D α has postve measure. In partcular, we can choose z Z \ α J D α. Snce J s countable and I s not, there s β I \ J. We now defne an element x 2 κ by declarng z(γ f γ α J A α, x(γ := 0 f γ A β, 1 otherwse. We clam that x Z E β. Indeed, we have x Z by (8 (bear n mnd that A α J A α. On the other hand, take any α < κ wth α β. If α J then z D α and so x D α as well. If α J, then x(γ = 1 for all γ A α and so x D α. It follows that x Z E β, as clamed. Therefore Z ( α I E α. Lemma 5.4. Let κ be a cardnal wth κ > c. Then there s a partton (C γ γ Γ of 2 κ nto sets of measure zero such that, whenever (Γ A A N s a partton of Γ, there s some A N such that γ Γ A C γ has outer measure 1. Proof. Let (A α α<κ be a partton of κ nto nfnte sets. Clearly, the E α s of Lemma 5.3 are parwse dsjont and have measure zero (snce A α s nfnte. We clam that the followng partton of 2 κ satsfes the desred property: C := { E α : α < κ } { {x} : x 2 κ \ E α }. α<κ β<α Indeed, let (C A A N be any partton of C. Snce κ > c, there s some A N such that C A contans uncountably many E α s. By Lemma 5.3, the outer measure of CA s 1, as requred. We can now state the aforementoned result: Theorem 5.5. Let κ be a cardnal wth κ > c. Then there s a McShane ntegrable functon f : 2 κ c 0 (Γ (for some set Γ whch s not MC-ntegrable.

13 THE MCSHANE INTEGRAL IN WEAKLY COMPACTLY GENERATED SPACES 13 Proof. By Lemmas 5.1, 5.2 and 5.4, there s a scalarly null functon f : 2 κ c 0 (Γ (for some set Γ whch s not MC-ntegrable. Snce f s Petts ntegrable, t s also McShane ntegrable [2]. In [1] t s proved that Petts and McShane ntegrablty are equvalent for X- valued functons defned on quas-radon probablty spaces whenever X s Hlbert generated (.e., there exst a Hlbert space Y and a lnear contnuous map T : Y X such that T (Y s dense n X. Clearly, every Hlbert generated space s WCG. Typcal examples of Hlbert generated spaces are the separable ones, c 0 (Γ (for any set Γ and L 1 (ν (for any probablty measure ν. Moreover, any super-reflexve space embeds nto a Hlbert generated space. For more nformaton on ths class of spaces, we refer the reader to [5, 6] and [14, Chapter 6]. In vew of our Theorem 5.5, we cannot replace McShane ntegrablty by MCntegrablty n the results of [1]. However, somethng can be sad for a partcular class of functons. The followng proposton s nspred n [1, Lemma 3.3]. Recall that a Markushevch bass of X s a famly {(x, x : I} X X such that x (x j = δ,j for every, j I, the lnear span of {x : I} s dense n X and {x : I} separates the ponts of X. Proposton 5.6. Suppose µ s atomless and X s a closed subspace of a Hlbert generated Banach space. Let {(x, x : I} be a Markushevch bass of X wth x B X for all I. Let ϕ : Ω I be a one-to-one functon and defne f : Ω X by f(t := x ϕ(t. Then f s scalarly null and MC-ntegrable. Proof. Fx ε > 0. Snce X embeds nto a Hlbert generated space, there s a partton I = m N I m such that (9 for all x B X and all m N, { I m : x (x > ε} m, see [6, Theorem 6] (cf. [14, Theorem 6.30]. In partcular, for each x B X, the set {t Ω : x f(t > ε} s countable (ϕ s one-to-one and so t has outer measure 0 (because µ s atomless. As ε > 0 s arbtrary, f s scalarly null. For each m N, defne Ω m := {t Ω : ϕ(t I m } and choose fntely many dsjont sets A 1,m,..., A N(m,m Σ wth µ(a n,m ε 2 m, n = 1,..., N(m, m and Ω m N(m n=1 A n,m. Set Ω n,m := Ω m A n,m for all m N and n = 1,..., N(m, so that (Ω n,m s a countable partton of Ω. Fx a fnte famly (E j of parwse dsjont elements of Σ wth E j A n(j,m(j and choose ponts t j Ω n(j,m(j. Fx x B X and set C := { I : x (x ε} and B m := { I m : x (x > ε} for all m N. We can wrte (10 µ(e j f(t j = µ(e j x + j C ϕ(t j= m N B m ϕ(t j= µ(e j x.

14 14 A. AVILÉS, G. PLEBANEK, AND J. RODRÍGUEZ On the one hand (11 x ( C ϕ(t j= µ(e j x µ ( C ϕ(t j= E j ε ε. On the other hand, for each m N and B m, we have ϕ(t E j= j A n,m for some n (here we use agan that ϕ s one-to-one and therefore (12 x ( ( µ(e j x µ E j ε 2 m m. ϕ(t j= ϕ(t j= From (9, (10, (11 and (12 t follows that x ( µ(e j f(t j 2ε. j As x B X s arbtrary, we have j µ(e jf(t j 2ε. Hence f s MCntegrable, wth MC-ntegral 0 X. References [1] R. Devlle and J. Rodríguez, Integraton n Hlbert generated Banach spaces, Israel J. Math. (to appear. [2] L. D Pazza and D. Press, When do McShane and Petts ntegrals concde?, Illnos J. Math. 47 (2003, no. 4, MR (2005a:28023 [3] J. Destel, Geometry of Banach spaces selected topcs, Sprnger-Verlag, Berln, 1975, Lecture Notes n Mathematcs, Vol MR (57 #1079 [4] J. Destel and J. J. Uhl, Jr., Vector measures, Amercan Mathematcal Socety, Provdence, R.I., 1977, Wth a foreword by B. J. Petts, Mathematcal Surveys, No. 15. MR (56 #12216 [5] M. Faban, G. Godefroy, P. Hájek, and V. Zzler, Hlbert-generated spaces, J. Funct. Anal. 200 (2003, no. 2, MR (2004b:46011 [6] M. Faban, G. Godefroy, V. Montesnos, and V. Zzler, Inner characterzatons of weakly compactly generated Banach spaces and ther relatves, J. Math. Anal. Appl. 297 (2004, no. 2, , Specal ssue dedcated to John Horváth. MR (2005g:46046 [7] K. Floret, Weakly compact sets, Lecture Notes n Mathematcs, vol. 801, Sprnger, Berln, 1980, Lectures held at S.U.N.Y., Buffalo, n Sprng MR (82b:46001 [8] D. H. Fremln, Problem DU, note of , avalable at [9] D. H. Fremln, Problem ET, note of , avalable at [10] D. H. Fremln, The generalzed McShane ntegral, Illnos J. Math. 39 (1995, no. 1, MR (95j:28008 [11] D. H. Fremln, Measure theory. Vol. 4, Torres Fremln, Colchester, 2006, Topologcal measure spaces. Part I, II, Corrected second prntng of the 2003 orgnal. MR [12] D. H. Fremln and J. Mendoza, On the ntegraton of vector-valued functons, Illnos J. Math. 38 (1994, no. 1, MR (94k:46083 [13] R. A. Gordon, The McShane ntegral of Banach-valued functons, Illnos J. Math. 34 (1990, no. 3, MR (91m:26010 [14] P. Hájek, V. Montesnos Santalucía, J. Vanderwerff, and V. Zzler, Borthogonal systems n Banach spaces, CMS Books n Mathematcs/Ouvrages de Mathématques de la SMC, 26, Sprnger, New York, MR (2008k:46002 [15] B. J. Petts, On ntegraton n vector spaces, Trans. Amer. Math. Soc. 44 (1938, no. 2, MR

15 THE MCSHANE INTEGRAL IN WEAKLY COMPACTLY GENERATED SPACES 15 [16] J. Rodríguez, On the equvalence of McShane and Petts ntegrablty n non-separable Banach spaces, J. Math. Anal. Appl. 341 (2008, no. 1, MR (2009b:46095 [17] J. Rodríguez, Some examples n vector ntegraton, Bull. Aust. Math. Soc. (2009, do: /s [18] M. Talagrand, Petts ntegral and measure theory, Mem. Amer. Math. Soc. 51 (1984, no. 307, x+224. MR (86j:46042 Departamento de Matemátcas, Facultad de Matemátcas, Unversdad de Murca, Espnardo (Murca, Span E-mal address: avleslo@um.es Mathematcal Insttute, Unversty of Wroc law, Pl. Grunwaldzk 2/4, Wroc- law, Poland E-mal address: grzes@math.un.wroc.pl Departamento de Matemátca Aplcada, Facultad de Informátca, Unversdad de Murca, Espnardo (Murca, Span E-mal address: joserr@um.es