November  2021, 4(4): 299-310. doi: 10.3934/mfc.2021020

A note on convergence results for varying interval valued multisubmeasures

1. 

University ''Alexandru Ioan Cuza'', Faculty of Mathematics, Bd. Carol I, No. 11, Iaşi, 700506, Romania

2. 

Petroleum-Gas University of Ploieşti, Department of Computer Science, Information Technology, Mathematics and Physics, Bd. Bucureşti, No. 39, Ploieşti 100680, Romania

3. 

Department of Mathematics and Computer Sciences, University of Perugia, 1, Via Vanvitelli, 06123 Perugia, Italy

*Corresponding author: Anna Rita Sambucini

Received  June 2021 Revised  August 2021 Published  November 2021 Early access  September 2021

Some limit theorems are presented for Riemann-Lebesgue integrals where the functions $ G_n $ and the measures $ M_n $ are interval valued and the convergence for the multisubmeasures is setwise. In particular sufficient conditions in order to obtain $ \int G_n dM_n \to \int G dM $ are given.

Citation: Anca Croitoru, Alina GavriluŢ, Alina Iosif, Anna Rita Sambucini. A note on convergence results for varying interval valued multisubmeasures. Mathematical Foundations of Computing, 2021, 4 (4) : 299-310. doi: 10.3934/mfc.2021020
References:
[1]

L. AngeloniJ. Appell and S. Reinwand, Some remarks on Vainikko integral operators in BV type spaces, Boll. Unione Mat. Ital., 13 (2020), 555-565.  doi: 10.1007/s40574-020-00248-3.

[2]

L. AngeloniD. CostarelliM. SeraciniG. Vinti and L. Zampogni, Variation diminishing-type properties for multivariate sampling Kantorovich operators, Boll. dell'Unione Matem. Ital., 13 (2020), 595-605.  doi: 10.1007/s40574-020-00256-3.

[3]

M. L. Avendaño-GarridoJ. R. Gabriel-ArgüellesL. Torres Quintana and J. González-Hernández, An approximation scheme for the Kantorovich-Rubinstein problem on compact spaces, J. Numer. Math., 26 (2018), 63-75.  doi: 10.1515/jnma-2017-0008.

[4]

C. BardaroA. Boccuto and I. Mantellini, A survey on recent results in Korovkin's approximation theory in modular spaces, Constr. Math. Anal., 4 (2021), 48-60.  doi: 10.33205/cma.804697.

[5]

C. Bardaro and I. Mantellini, On convergence properties for a class of Kantorovich discrete operators, Numer. Funct. Anal. Optim., 33 (2012), 374-396.  doi: 10.1080/01630563.2011.652270.

[6]

A. Boccuto and D. Candeloro, Integral and ideals in Riesz spaces, Inform. Sci., 179 (2009) 2891–2902. doi: 10.1016/j.ins.2008.11.001.

[7]

D. CandeloroA. CroitoruA. GavriluŢA. Iosif and A. R. Sambucini, Properties of the Riemann-Lebesgue integrability in the non-additive case, Rend. Circ. Mat. Palermo, Serie 2, 69 (2020), 577-589.  doi: 10.1007/s12215-019-00419-y.

[8]

D. CandeloroA. CroitoruA. GavriluŢ and A. R. Sambucini, An extension of the Birkhoff integrability for multifunctions, Mediterr. J. Math., 13 (2016), 2551-2575.  doi: 10.1007/s00009-015-0639-7.

[9]

D. CandeloroL. D. PiazzaK. Musial and A. R. Sambucini, Multi-integrals of finite variation, Boll. Unione Mat. Ital., 13 (2020), 459-468.  doi: 10.1007/s40574-020-00217-w.

[10]

D. CaponettiV. Marraffa and K. Naralenkov, On the integration of Riemann-measurable vector-valued functions, Monatsh. Math., 182 (2017), 513-536.  doi: 10.1007/s00605-016-0923-z.

[11]

F. Cluni, D. Costarelli, V. Gusella and G. Vinti, Reliability increase of masonry characteristics estimation by a sampling algorithm applied to thermographic digital images, Probabilistic Engineering Mechanics, 60 (2020) 103022. doi: 10.1016/j.probengmech.2020.103022.

[12]

G. ColettiD. Petturiti and B. Vantaggi, Models for pessimistic or optimistic decisions under different uncertain scenarios, Internat. J. Approx. Reason., 105 (2019), 305-326.  doi: 10.1016/j.ijar.2018.12.005.

[13]

D. CostarelliA. CroitoruA. GavriluŢA. Iosif and A. R. Sambucini, The Riemann-Lebesgue integral of interval-valued multifunctions, Mathematics, 8 (2020), 1-17.  doi: 10.3390/math8122250.

[14]

D. CostarelliM. Seracini and G. Vinti, A segmentation procedure of the pervious area of the aorta artery from CT images without contrast medium, Math. Methods Appl. Sci., 43 (2020), 114-133.  doi: 10.1002/mma.5838.

[15]

D. CostarelliM. Seracini and G. Vinti, A comparison between the sampling Kantorovich algorithm for digital image processing with some interpolation and quasi-interpolation methods, Appl. Math. Comput., 374 (2020), 125046.  doi: 10.1016/j.amc.2020.125046.

[16]

A. P. Dempster, Upper and lower probabilities induced by a multivalued mapping, Ann. Math. Statist., 38 (1967), 325-339.  doi: 10.1214/aoms/1177698950.

[17]

N. Dinculeanu, Vector measures, international series of monographs in pure and applied mathematics, Pergamon Press, Oxford-New York-Toronto, Ont.; VEB Deutscher Verlag der Wissenschaften, Berlin, 95 (1967), 432pp.

[18]

L. Drewnowski, Topological rings of sets, continuous set functions, integration, Ⅰ, Ⅱ, Bull. Acad. Polon. Sci. Ser. Math. Astron. Phys., 20 (1972), 277-286. 

[19]

S. G. Gal, On a Choquet-Stieltjes type integral on intervals, Math. Slovaca, 69 (2019), 801-814.  doi: 10.1515/ms-2017-0269.

[20]

A. GavriluŢ, Remarks on monotone interval valued set multifunctions, Inform. Sci., 259 (2014), 225-230.  doi: 10.1016/j.ins.2013.08.032.

[21]

C. Guo and D. Zhang, On set-valued fuzzy measures, Inform. Sci., 160 (2004), 13-25.  doi: 10.1016/j.ins.2003.07.006.

[22]

O. Hernandez-Lerma and J. B. Lasserre, Fatou's lemma and lebesgue's convergence theorem for measures, J. Appl. Math. Stoch. Anal., 13 (2000), 137-146.  doi: 10.1155/S1048953300000150.

[23]

A. Jurio, D. Paternain, C. Lopez-Molina, H. Bustince, R. Mesiar and G. Beliakov, A construction method of interval-valued fuzzy sets for image processing, 2011 IEEE Symposium on Advances in Type-2 Fuzzy Logic Systems, (2011). doi: 10.1109/T2FUZZ.2011.5949554.

[24]

V. M. KadetsB. ShumyatskiyR. ShvidkoyL. M. Tseytlin and K. Zheltukhin, Some remarks on vector-valued integration, Mat. Fiz. Anal. Geom., 9 (2002), 48-65. 

[25]

V. M. Kadets and L. M. Tseytlin, On integration of non-integrable vector-valued functions, Mat. Fiz. Anal. Geom., 7 (2000), 49-65. 

[26]

S. B. Kaliaj, A Kannan-type fxed point theorem for multivalued mappings with application, J. Anal., 27 (2019), 837-849.  doi: 10.1007/s41478-018-0135-0.

[27]

E. Z. Kotonaj Kallushi, Birkhoff integral in quasi-Banach spaces, European Scientific Journal, 8 (2018), 188-196. 

[28]

J. B. Lasserre, On the setwise convergence of sequences of measures, J. Appl. Math. and Stoch. Anal., 10 (1997), 131-136.  doi: 10.1155/S1048953397000166.

[29]

D. La Torre and F. Mendivil, Minkowski-additive multimeasures, monotonicity and self-similarity, Image Anal. Stereol., 30 (2011), 135-142.  doi: 10.5566/ias.v30.p135-142.

[30]

C. Lopez-MolinaB. De BaetsE. Barrenechea and H. Bustince, Edge detection on interval-valued images, Advances in Intelligent and Soft Computing, 107 (2011), 325-337.  doi: 10.1007/978-3-642-24001-0_30.

[31]

L. D. Piazza, V. Marraffa, K. Musiał and A. R. Sambucini, Convergence for Varying Measures, Work in Progress, 2021.

[32]

L. D. PiazzaV. Marraffa and B. Satco, Measure differential inclusions: Existence results and minimum problems, Set-Valued Var. Anal., 29 (2021), 361-382.  doi: 10.1007/s11228-020-00559-9.

[33]

L. Di Piazza and K. Musiał, Decompositions of weakly compact valued integrable multifunctions, Mathematics, 8 (2020), 863.  doi: 10.3390/math8060863.

[34]

L. D. PiazzaV. Marraffa and B. Satco, Measure differential inclusions: Existence results and minimum problems, Set-Valued Var. Anal., 29 (2021), 361-382.  doi: 10.1007/s11228-020-00559-9.

[35]

E. Pap, Pseudo-additive measures and their applications, Handbook of Measure Theory, 2 (2002), 1403-1465.  doi: 10.1016/B978-044450263-6/50036-1.

[36]

E. PapA. Iosif and A. GavriluŢ, Integrability of an interval-valued multifunction with respect to an interval-valued set multifunction, Iranian Journal of Fuzzy Systems, 15 (2018), 47-63. 

[37]

H. Román-FloresY. Chalco-Cano and W. A. Lodwick, Some integral inequalities for interval-valued functions, Comp. Appl. Math., 37 (2018), 1306-1318.  doi: 10.1007/s40314-016-0396-7.

[38]

C. Stamate and A. Croitoru, The general Pettis-Sugeno integral of vector multifunctions relative to a vector fuzzy multimeasure, Fuzzy Sets and Systems, 327 (2017), 123-136.  doi: 10.1016/j.fss.2017.07.007.

[39]

V. Torra, Use and applications of non-additive measures and integrals, Studies in Fuzziness and Soft Computing, 310 (2014), 1-33.  doi: 10.1007/978-3-319-03155-2_2.

[40]

K. Weichselberger, The theory of interval-probability as a unifying concept for uncertainty, Int. J. Approx. Reason., 24 (2000), 149-170.  doi: 10.1016/S0888-613X(00)00032-3.

[41]

D. Zhang and C. Guo, On the convergence of sequences of fuzzy measures and generalized convergence theorems of fuzzy integrals, Fuzzy Sets Systems, 72 (1995), 349-356.  doi: 10.1016/0165-0114(94)00290-N.

[42]

Y. ZhouC. Zhang and Z. Zhang, An efficient fractal image coding algorithm using unified feature and DCT, Chaos, Solitons & Fractals, 39 (2009), 1823-1830. 

show all references

References:
[1]

L. AngeloniJ. Appell and S. Reinwand, Some remarks on Vainikko integral operators in BV type spaces, Boll. Unione Mat. Ital., 13 (2020), 555-565.  doi: 10.1007/s40574-020-00248-3.

[2]

L. AngeloniD. CostarelliM. SeraciniG. Vinti and L. Zampogni, Variation diminishing-type properties for multivariate sampling Kantorovich operators, Boll. dell'Unione Matem. Ital., 13 (2020), 595-605.  doi: 10.1007/s40574-020-00256-3.

[3]

M. L. Avendaño-GarridoJ. R. Gabriel-ArgüellesL. Torres Quintana and J. González-Hernández, An approximation scheme for the Kantorovich-Rubinstein problem on compact spaces, J. Numer. Math., 26 (2018), 63-75.  doi: 10.1515/jnma-2017-0008.

[4]

C. BardaroA. Boccuto and I. Mantellini, A survey on recent results in Korovkin's approximation theory in modular spaces, Constr. Math. Anal., 4 (2021), 48-60.  doi: 10.33205/cma.804697.

[5]

C. Bardaro and I. Mantellini, On convergence properties for a class of Kantorovich discrete operators, Numer. Funct. Anal. Optim., 33 (2012), 374-396.  doi: 10.1080/01630563.2011.652270.

[6]

A. Boccuto and D. Candeloro, Integral and ideals in Riesz spaces, Inform. Sci., 179 (2009) 2891–2902. doi: 10.1016/j.ins.2008.11.001.

[7]

D. CandeloroA. CroitoruA. GavriluŢA. Iosif and A. R. Sambucini, Properties of the Riemann-Lebesgue integrability in the non-additive case, Rend. Circ. Mat. Palermo, Serie 2, 69 (2020), 577-589.  doi: 10.1007/s12215-019-00419-y.

[8]

D. CandeloroA. CroitoruA. GavriluŢ and A. R. Sambucini, An extension of the Birkhoff integrability for multifunctions, Mediterr. J. Math., 13 (2016), 2551-2575.  doi: 10.1007/s00009-015-0639-7.

[9]

D. CandeloroL. D. PiazzaK. Musial and A. R. Sambucini, Multi-integrals of finite variation, Boll. Unione Mat. Ital., 13 (2020), 459-468.  doi: 10.1007/s40574-020-00217-w.

[10]

D. CaponettiV. Marraffa and K. Naralenkov, On the integration of Riemann-measurable vector-valued functions, Monatsh. Math., 182 (2017), 513-536.  doi: 10.1007/s00605-016-0923-z.

[11]

F. Cluni, D. Costarelli, V. Gusella and G. Vinti, Reliability increase of masonry characteristics estimation by a sampling algorithm applied to thermographic digital images, Probabilistic Engineering Mechanics, 60 (2020) 103022. doi: 10.1016/j.probengmech.2020.103022.

[12]

G. ColettiD. Petturiti and B. Vantaggi, Models for pessimistic or optimistic decisions under different uncertain scenarios, Internat. J. Approx. Reason., 105 (2019), 305-326.  doi: 10.1016/j.ijar.2018.12.005.

[13]

D. CostarelliA. CroitoruA. GavriluŢA. Iosif and A. R. Sambucini, The Riemann-Lebesgue integral of interval-valued multifunctions, Mathematics, 8 (2020), 1-17.  doi: 10.3390/math8122250.

[14]

D. CostarelliM. Seracini and G. Vinti, A segmentation procedure of the pervious area of the aorta artery from CT images without contrast medium, Math. Methods Appl. Sci., 43 (2020), 114-133.  doi: 10.1002/mma.5838.

[15]

D. CostarelliM. Seracini and G. Vinti, A comparison between the sampling Kantorovich algorithm for digital image processing with some interpolation and quasi-interpolation methods, Appl. Math. Comput., 374 (2020), 125046.  doi: 10.1016/j.amc.2020.125046.

[16]

A. P. Dempster, Upper and lower probabilities induced by a multivalued mapping, Ann. Math. Statist., 38 (1967), 325-339.  doi: 10.1214/aoms/1177698950.

[17]

N. Dinculeanu, Vector measures, international series of monographs in pure and applied mathematics, Pergamon Press, Oxford-New York-Toronto, Ont.; VEB Deutscher Verlag der Wissenschaften, Berlin, 95 (1967), 432pp.

[18]

L. Drewnowski, Topological rings of sets, continuous set functions, integration, Ⅰ, Ⅱ, Bull. Acad. Polon. Sci. Ser. Math. Astron. Phys., 20 (1972), 277-286. 

[19]

S. G. Gal, On a Choquet-Stieltjes type integral on intervals, Math. Slovaca, 69 (2019), 801-814.  doi: 10.1515/ms-2017-0269.

[20]

A. GavriluŢ, Remarks on monotone interval valued set multifunctions, Inform. Sci., 259 (2014), 225-230.  doi: 10.1016/j.ins.2013.08.032.

[21]

C. Guo and D. Zhang, On set-valued fuzzy measures, Inform. Sci., 160 (2004), 13-25.  doi: 10.1016/j.ins.2003.07.006.

[22]

O. Hernandez-Lerma and J. B. Lasserre, Fatou's lemma and lebesgue's convergence theorem for measures, J. Appl. Math. Stoch. Anal., 13 (2000), 137-146.  doi: 10.1155/S1048953300000150.

[23]

A. Jurio, D. Paternain, C. Lopez-Molina, H. Bustince, R. Mesiar and G. Beliakov, A construction method of interval-valued fuzzy sets for image processing, 2011 IEEE Symposium on Advances in Type-2 Fuzzy Logic Systems, (2011). doi: 10.1109/T2FUZZ.2011.5949554.

[24]

V. M. KadetsB. ShumyatskiyR. ShvidkoyL. M. Tseytlin and K. Zheltukhin, Some remarks on vector-valued integration, Mat. Fiz. Anal. Geom., 9 (2002), 48-65. 

[25]

V. M. Kadets and L. M. Tseytlin, On integration of non-integrable vector-valued functions, Mat. Fiz. Anal. Geom., 7 (2000), 49-65. 

[26]

S. B. Kaliaj, A Kannan-type fxed point theorem for multivalued mappings with application, J. Anal., 27 (2019), 837-849.  doi: 10.1007/s41478-018-0135-0.

[27]

E. Z. Kotonaj Kallushi, Birkhoff integral in quasi-Banach spaces, European Scientific Journal, 8 (2018), 188-196. 

[28]

J. B. Lasserre, On the setwise convergence of sequences of measures, J. Appl. Math. and Stoch. Anal., 10 (1997), 131-136.  doi: 10.1155/S1048953397000166.

[29]

D. La Torre and F. Mendivil, Minkowski-additive multimeasures, monotonicity and self-similarity, Image Anal. Stereol., 30 (2011), 135-142.  doi: 10.5566/ias.v30.p135-142.

[30]

C. Lopez-MolinaB. De BaetsE. Barrenechea and H. Bustince, Edge detection on interval-valued images, Advances in Intelligent and Soft Computing, 107 (2011), 325-337.  doi: 10.1007/978-3-642-24001-0_30.

[31]

L. D. Piazza, V. Marraffa, K. Musiał and A. R. Sambucini, Convergence for Varying Measures, Work in Progress, 2021.

[32]

L. D. PiazzaV. Marraffa and B. Satco, Measure differential inclusions: Existence results and minimum problems, Set-Valued Var. Anal., 29 (2021), 361-382.  doi: 10.1007/s11228-020-00559-9.

[33]

L. Di Piazza and K. Musiał, Decompositions of weakly compact valued integrable multifunctions, Mathematics, 8 (2020), 863.  doi: 10.3390/math8060863.

[34]

L. D. PiazzaV. Marraffa and B. Satco, Measure differential inclusions: Existence results and minimum problems, Set-Valued Var. Anal., 29 (2021), 361-382.  doi: 10.1007/s11228-020-00559-9.

[35]

E. Pap, Pseudo-additive measures and their applications, Handbook of Measure Theory, 2 (2002), 1403-1465.  doi: 10.1016/B978-044450263-6/50036-1.

[36]

E. PapA. Iosif and A. GavriluŢ, Integrability of an interval-valued multifunction with respect to an interval-valued set multifunction, Iranian Journal of Fuzzy Systems, 15 (2018), 47-63. 

[37]

H. Román-FloresY. Chalco-Cano and W. A. Lodwick, Some integral inequalities for interval-valued functions, Comp. Appl. Math., 37 (2018), 1306-1318.  doi: 10.1007/s40314-016-0396-7.

[38]

C. Stamate and A. Croitoru, The general Pettis-Sugeno integral of vector multifunctions relative to a vector fuzzy multimeasure, Fuzzy Sets and Systems, 327 (2017), 123-136.  doi: 10.1016/j.fss.2017.07.007.

[39]

V. Torra, Use and applications of non-additive measures and integrals, Studies in Fuzziness and Soft Computing, 310 (2014), 1-33.  doi: 10.1007/978-3-319-03155-2_2.

[40]

K. Weichselberger, The theory of interval-probability as a unifying concept for uncertainty, Int. J. Approx. Reason., 24 (2000), 149-170.  doi: 10.1016/S0888-613X(00)00032-3.

[41]

D. Zhang and C. Guo, On the convergence of sequences of fuzzy measures and generalized convergence theorems of fuzzy integrals, Fuzzy Sets Systems, 72 (1995), 349-356.  doi: 10.1016/0165-0114(94)00290-N.

[42]

Y. ZhouC. Zhang and Z. Zhang, An efficient fractal image coding algorithm using unified feature and DCT, Chaos, Solitons & Fractals, 39 (2009), 1823-1830. 

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