Semicontinuous filter limits of nets of lattice groupvalued

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1 Semicontinuous ilter limits o nets o lattice grouvalued unctions THEMATIC UNIT: MATHEMATICS AND APPLICATIONS A Boccuto, Diartimento di Matematica e Inormatica, via Vanvitelli, I- 623 Perugia, Italy, antonioboccuto@unigit, boccuto@yahooit X Dimitriou (corresonding author, Deartment o Mathematics, University o Athens, Paneistimioolis, Athens 5784, Greece, enoon@gmailcom, deno@mathuoagr ABSTRACT: Some necessary and suicient conditions or semicontinuity o the limit unction o a ointwise convergent net o lattice grou-valued unctions resect to ilter convergence are given In this ramework we consider some kinds o ilter ehaustiveness Furthermore, we ose some oen roblems Introduction In this aer we deal the roblem o inding some conditions or semicontinuity o the limit o a ointwise convergent net o unctions, deined on toological saces and taking values in lattice grous, and continue the research started in [4] We use ilters and achieve some characteriation in terms o seciic eatures o unctions nets, etending some earlier results roved in [2] and [7] For a related literature, see [ -7] Finally we ose some oen roblems 2 The main results Deinitions 2 (a An abelian artially ordered grou L = ( L, +, neutral element is called a lattice grou (shortly, (l -grou i it is a lattice

2 and or every a, b, c L a b we get a + c b + c A meaningul eamle o an (l -grou, widely studied in the literature, is the vector lattice R = L ([,], B, ν o all ν -measurable unctions identiication u to ν -null sets, where B is the σ -algebra o all Borel subsets o [,] and ν is the Lebesgue measure Observe that, in R, order convergence coincides almost everywhere convergence, which does not have a toological nature (see also [4] (b An (O -sequence is a decreasing sequence in R,whose inimum is For the main roerties o ilters, (O -sequences and (ilter order convergence in lattice grous we reer to [4] Deinitions 22 (a Let ( Λ, be any directed set A ilter o Λ is said to be (Λ -ree i it contains the sets o the tye { ζ Λ : ζ } or every Λ When Λ = N, some eamles o (N -ree ilters are the ilter F coin o those subsets o N whose comlements are inite sets and the ilter F st o all subsets o N having asymtotic density one (see also [4] (b Let X be a Hausdor toological sace A unction : X R is uer semicontinuous at a oint X i there is an (O -sequence (σ (deending on such that or each N there is a neighborhood U o X ( ( whenever U We say that R is lower X semicontinuous at i is uer semicontinuous at A unction R is continuous at i it is both uer and lower semicontinuous at I the (O -sequence (σ can be chosen indeendently o X, then we say that is globally uer semicontinuous (res globally lower semicontinuous on X X A unction R is globally continuous on X i it is both globally uer and globally lower semicontinuous on X (c Let X We say that a net : X R, Λ, is F -uer ehaustive at i there is an (O -sequence (σ such that or any N there eist a neighborhood U o and a set F F such that or each F

3 and U we have ( ( We say that ( is F -lower ehaustive at i ( is F -uer ehaustive at (d A net : X R, Λ, is weakly F-uer ehaustive at i there is an (O -sequence (σ such that or each N there is a neighborhood U o such that or every U there is F F ( ( whenever F We say that ( is weakly F -lower ehaustive at i ( is weakly F -uer ehaustive at (e We say that : X R, Λ, is weakly F -uer (lower ehaustive on X i it is weakly F-uer (lower ehaustive at every X resect to a single (O -sequence, indeendent o X ( We say that ( is (weakly F -ehaustive at (res on X i it is both (weaklyf -uer and (weaklyf -lower ehaustive at (res on X (g We say that the net : X R, Λ, is F -almost below a unction : X R around a oint X i there eists an (O -sequence (σ such that or every N there is a neighborhood U o such that or each U there is a set F F ( ( whenever F We say that ( is F -almost above around i ( is F -almost below around (see also [7, Deinition 33], when Λ = N and F =Fcoin or F =F st (h A net ( is ( RO F -uer convergent to i there is an (O - sequence (σ such that or each N and X there is a set F F ( + ( σ whenever F We say that ( is ( ROF -lower convergent to i ( is ( F RO -uer convergent to, and that ( is ( F RO -convergent to i it is both ( RO F -uer and ( ROF - lower convergent to Remarks 23 (a Note that, when R = R, ( ROF -convergence coincides the usual ointwise ilter convergence, and the (global (semicontinuity coincides the ordinary (semicontinuity Moreover, when

4 F =F coin, ilter (order convergence coincides usual (order convergence (see also [4] (b Observe that, contrarily to ( ROF -convergence, it may haen that a unction net is ( ROF -uer (lower convergent to more than a unction Indeed, i h ( = or any X and Λ, then o course (h ( ROF - converges (only to the identically unction, but it is easy to see that, or each unction h (res, (h is ( ROF -uer (res lower convergent to h Eamle 24 Let Λ = ( Λ, =],+ [ be directed the usual order, X = [,] be endowed the usual distance, ν be the Lebesgue measure on X, and R be the sace o all bounded ν -measurable unctions on X, identiication u to ν -null sets, where order convergence is almost everywhere convergence dominated by an element o R, and so is not toological (see also [4] For each real number a, let a be the unction which associates to every element X the constant a Let F be any ied (Λ -ree ilter o Λ and, g : X R, Λ, be deined by, i [,], ( =, i =, g ( = (, i (,],, It is easy to check that ( and (g ( ROF -converge to resect to the (O -sequence σ =, N, and thus rom Theorem 25 it will ollow that they are both weakly uer and weak lower F -ehaustive on [,] On the other hand, it is not diicult to see that, i (σ is any (O - sequence in R, then there is a ositive real number M σ ( M or every N and X In corresondence M, or each set F F and or every neighborhood U o there is an element F, > M and

5 U So we get ( =, and hence ( σ From this it ( (res ollows that (g is not F -uer (res F -lower ehaustive at, and hence not F -ehaustive at Moreover, by construction, it is easy to check that ( (res (g is F -lower (res F -uer ehaustive at The ollowing result etends [7, Corollary 37] to the contet o unction nets, ree ilters and lattice grous Theorem 25 Let F be a (Λ -ree ilter o Λ, X be a Hausdor toological sace, : X R, Λ, be a net o unctions, ( ROF - convergent to : X R resect to a single (O -sequence ( σ, and X be a ied oint Then the ollowing are equivalent: (i ( is weakly F -uer ehaustive at ; (ii is uer semicontinuous at ; (iii ( is F -almost below around Proo: ( i ( ii Let (σ be an (O -sequence associated weak F -uer ehaustiveness o ( at, and ick N By (i there eists a neighborhood U o, related to weak F -uer ehaustiveness For each U there is a set F = F ( F ( ( or all F Moreover, thanks to ( ROF -convergence resect to the (O -sequence σ, or every, and there eists a set 2 ( σ and ( ( + ( ( + F = F (,, F 2 σ whenever F2 Thus or every F F 2 and U we have ( ( 2σ So we get (ii ( ii ( iii Let ( σ and (σ be two (O -sequences related to ( ROF -convergence and uer semicontinuity o at resectively Fi arbitrarily N By uer semicontinuity o at there is a neighborhood U o ( ( whenever U By ( ROF -convergence, or each U there is a set F F ( ( or all F Thus ( ( or any,

6 F, getting (iii ( iii ( i Let arbitrarily (σ be an (O -sequence, according to (iii Choose N By ( ROF -convergence, there is a set F F ( ( or every F By (iii, we ind a neighborhood U o such that or every U there is a set F F ( ( ( ( or each F Thus whenever U and F F F, getting (i This ends the roo Analogously as in Theorem 25, it is ossible to rove the ollowing Theorem 26 Let F, Λ, X be as in Theorem 25, : X R, Λ, be a unction net, ( ROF -convergent to : X R, and X Then the ollowing are equivalent: (i ( is weakly F -lower ehaustive at ; (ii is lower semicontinuous at ; (iii ( is F -almost above around Deinition 27 A net ( is said to be ( F - s T -uer convergent to i there is an (O -sequence (σ such that ( ( ROF -converges to resect to (σ and or each N and X there is a set F F such that or every F there is a neighborhood U o such that ( ( whenever U We say that ( is ( F - s T -lower convergent to i ( is ( F - s T -uer convergent to Remark 28 In general, even when R = R, = N Λ and F= F coin, the ointwise limit o a convergent sequence o continuous unctions is not uer n semicontinous: indeed, it is enough to take n( =, [,], n N It is easy to see that the limit unction, deined by setting ( = i [,[ and ( = i =, is not uer semicontinuous, though n is continuous or each n From this and Theorem 22 we will deduce that Deinition 27 is strictly stronger than Deinition 22 (h Theorem 29 Let X be a Hausdor toological sace, : X R be

7 globally uer semicontinuous on X, : X R, Λ, be a net o unctions globally lower semicontinuous on X resect to a single (O -sequence indeendent o, and ( ROF -uer convergent to Suose that (i ( is F -almost below around every oint X resect to a single (O -sequence, indeendent o Then ( is ( F - s T -uer convergent to Proo: Choose arbitrarily X Since ( is ( ROF -uer convergent to, there eists an (O -sequence ( σ, such that or every N there is a set F F σ or every F By global ( ( + lower semicontinuity o the s, there is an (O -sequence (σ such that or each N, Λ there eists a neighborhood U o ( ( σ whenever U By global uer semicontinuity o there is an (O -sequence (τ such that or every N there is a neighborhood V o ( ( + τ whenever V Since ( is F -almost below around, there eists an (O -sequence (ζ (indeendent o the roerty that or every N there eists a neighborhood W o such that or every W there is a set F F, ( ( ζ Set Z := U V and E := F F : note that E F We get + ( ( + τ ( + τ ( + τ F; ( ( + ζ ; ( ( + ζ or any Z, or every E From this the assertion ollows A consequence o Theorems 25 and 29 is the ollowing Corollary 2 Let X be a Hausdor toological sace, : X R be globally uer semicontinuous on X, : X R, Λ, be a net o unctions, globally lower semicontinuous on X resect to a single (O - sequence indeendent o, and ( ROF -convergent to Then ( is ( F - s T -uer convergent to

8 Remark 2 has been eliminated We now rove the ollowing version o the converse o Corollary 2 Theorem 2 Let X, R and be as in Corollary 2, : X R, Λ, be a net o unctions, globally continuous on X resect to a single s (O -sequence indeendent o, ( ROF -convergent to and ( F - T - uer convergent to Then ( and satisy condition (i o Theorem 29 Beore giving Theorem 2, we rove the ollowing Theorem 22 Let : X R, Λ, be a net o unctions, globally continuous on X resect to a single (O -sequence indeendent o, and : X R be a unction Then the ollowing are equivalent: (i ( is ( F - s T -uer convergent to ; (ii ( is ( F RO -convergent to and there is an (O -sequence in R such that or each nonemty comact subset C X, N and F F there are a inite set {, 2, K, k} F and an oen set U C, such that or every U there is [,k] ( ( ; (iii ( is ( ROF -convergent to and is globally uer semicontinuous on X Proo: ( i ( ii Let (σ be an (O -sequence related ( F - s T -uer convergence o corresondence each ( to Pick arbitrarily N and F F In X and there eists a set F F, such that or each F there is an oen neighborhood U = U, o ( ( σ or all U Put E := F F : since E F, we get that + E Let be any ied element o E and set nonemty comact subset o X There is a inite subcover C, and so it is not diicult to check that the sets V U, { := Let C be any V, V, K, } o V U k = V 2 k := V and {,, K, } are an oen set containing C and a inite subset o F 2 k

9 resectively, satisying (ii ( ii ( iii Let (ξ be an (O -sequence satisying an (O -sequence, such that ( ( ROF (σ Let (τ be an (ii and (σ be -converges to resect to (O -sequence, associated global continuity o the s and indeendent o Pick arbitrarily X and N By ( ROF - convergence there eists F F (deending on and ( ( σ or each F By (ii, in corresondence the comact set {} there eist a inite set {, 2, K, k} F and an oen neighborhood U o such that or every U there is [,k] ( ( ξ Since is globally continuous resect to the (O -sequence (τ, in ( corresondence and there is a neighborhood W o ( ( ( τ or every U W, and so we get ( ( = ( ( ( + ( ( ( + ( ( ( ( ( ( + ( ( + ( ( k ( ξ or every + τ U IW = This roves global uer semicontinuity o ( iii ( i See Corollary 2 Proo o Theorem 2: It is an immediate consequence o ( i ( iii o Theorem 22 and ( ii ( iii o Theorem 25 Remark 23 (a Observe that the imlication ( i ( ii o Theorem 22 holds out assuming any continuity or regularity hyothesis on the s, and the imlication ( ii ( iii holds even i the s are globally uer semicontinuous resect to a single (O -sequence indeendent o (see also [2, Proosition 37] (b Observe that, i ( is a net o globally continuous unctions, then

10 to get contemorarily ( ROF -convergence and semicontinuity o the limit s unction, ( F - T -uer convergence, condition (i o Theorem 29 and condition (ii o Theorem 22 are equivalent When the involved unctions are not continuous, even when R = R, Λ = N and F = F coin, in general ointwise s convergence and continuity o the limit unction does not imly ( F - T - uer convergence, as it is shown in [2, Eamle 33] Acknowledgement: Our thanks to the reerees or their remarks which imroved the eosition o the aer Oen roblems: (a Prove similar results in other contets and/or resect to dierent kinds o convergence (b Investigate some roerties related semicontinuity or unctions/measures values in abstract structures Reerences [] G Beer, The Aleandro roerty and the reservation o strong uniorm continuity, Al Gen Toology (2 (2, 7-33 [2] G Beer, Semicontinuous limits o nets o continuous unctions, Math Program, Ser B 39 (23, 7-79 [3] A Boccuto, P Das, X Dimitriou and N Paanastasiou, Ideal ehaustiveness, weak convergence and weak comactness in Banach saces, Real Anal Echange 37 (2 (22, [4] A Boccuto and X Dimitriou, Convergence Theorems or Lattice Grou-Valued Measures, Bentham Science Publ, U A E, 24, in ress [5] A Boccuto and X Dimitriou, Strong uniorm continuity and ilter ehaustiveness o nets o cone metric sace-valued unctions (24, vixra:4744 [6] D Candeloro and A R Sambucini, Filter convergence and decomositions or vector lattice-valued measures, Mediterranean J Math (24, to aear doi: 7/s [7] A Caserta, Decomosition o toologies which characterie the uer and lower semicontinuous limits o unctions, Abstr Al Anal, Article ID (2, 9 ages

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