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Volume 8
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Article

The Riemann-Lebesgue Integral of Interval-Valued Multifunctions

by
Danilo Costarelli
1,
Anca Croitoru
2,
Alina Gavriluţ
2,
Alina Iosif
3 and
Anna Rita Sambucini
1,*
1
Department of Mathematics and Computer Sciences, University of Perugia, 1, Via Vanvitelli, 06123 Perugia, Italy
2
Faculty of Mathematics, University Alexandru Ioan Cuza, Bd. Carol I, No. 11, 700506 Iaşi, Romania
3
Department of Computer Science, Information Technology, Mathematics and Physics, Petroleum-Gas University of Ploieşti, Bd. Bucureşti, No. 39, 100680 Ploieşti, Romania
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(12), 2250; https://doi.org/10.3390/math8122250
Submission received: 10 November 2020 / Revised: 14 December 2020 / Accepted: 16 December 2020 / Published: 20 December 2020
(This article belongs to the Special Issue Set-Valued Analysis)
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Abstract

We study Riemann-Lebesgue integrability for interval-valued multifunctions relative to an interval-valued set multifunction. Some classic properties of the RL integral, such as monotonicity, order continuity, bounded variation, convergence are obtained. An application of interval-valued multifunctions to image processing is given for the purpose of illustration; an example is given in case of fractal image coding for image compression, and for edge detection algorithm. In these contexts, the image modelization as an interval valued multifunction is crucial since allows to take into account the presence of quantization errors (such as the so-called round-off error) in the discretization process of a real world analogue visual signal into a digital discrete one.
Keywords: Riemann-Lebesgue integral; interval valued (set) multifunction; non-additive set function; image processing Riemann-Lebesgue integral; interval valued (set) multifunction; non-additive set function; image processing

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MDPI and ACS Style

Costarelli, D.; Croitoru, A.; Gavriluţ, A.; Iosif, A.; Sambucini, A.R. The Riemann-Lebesgue Integral of Interval-Valued Multifunctions. Mathematics 2020, 8, 2250. https://doi.org/10.3390/math8122250

AMA Style

Costarelli D, Croitoru A, Gavriluţ A, Iosif A, Sambucini AR. The Riemann-Lebesgue Integral of Interval-Valued Multifunctions. Mathematics. 2020; 8(12):2250. https://doi.org/10.3390/math8122250

Chicago/Turabian Style

Costarelli, Danilo, Anca Croitoru, Alina Gavriluţ, Alina Iosif, and Anna Rita Sambucini. 2020. "The Riemann-Lebesgue Integral of Interval-Valued Multifunctions" Mathematics 8, no. 12: 2250. https://doi.org/10.3390/math8122250

APA Style

Costarelli, D., Croitoru, A., Gavriluţ, A., Iosif, A., & Sambucini, A. R. (2020). The Riemann-Lebesgue Integral of Interval-Valued Multifunctions. Mathematics, 8(12), 2250. https://doi.org/10.3390/math8122250

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