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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[25]

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

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[25]

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

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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