Convergence for varying measures in the topological case

Piazza, L. Di; Marraffa, Valeria; Musiał, Kazimierz; Sambucini, Anna Rita

doi:10.1007/s10231-023-01353-8

Cited by 3 publications

(8 citation statements)

References 23 publications

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Section: Discussionmentioning

confidence: 99%

“…References [4,5] also worked with convergence for sequences of measures. Although [4] supposed that their functions were defined on Hausdorff topological spaces (see the first paragraph of Section 2 in [4]), they were taking values as scalars-their proofs were carried out by using absolute values. Some results of Reference [5] are for "vector-valued functions" (see Page 14 "Section 3.1 The vector case for integrals" in [5])-the proofs were carried out by using a norm-as a Banach space has.…”

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

“…References [4][5][6][7] all required that measures are bounded and converge "set-wisely". Some results in [4] require the sequence to converge in value and/or uniformly and absolutely continuously, while similar results in [6] require the measures to be finitevalued and/or increasing. Our results, however, only require "a sequence of bounded measures"-weaker conditions compared with [4][5][6][7]-in which both sequences of functions and measures are considered.…”

Section: Discussionmentioning

confidence: 99%

“…There is a rich bibliography concerning the convergence of sequences of measures, besides the above-mentioned studies [1-3]; see, for example, [4][5][6][7][8].…”

Section: Introduction and Terminologymentioning

confidence: 99%

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µ-Integrable Functions and Weak Convergence of Probability Measures in Complete Paranormed Spaces

Zeng

2024

Mathematics

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This paper works with functions defined in metric spaces and takes values in complete paranormed vector spaces or in Banach spaces, and proves some necessary and sufficient conditions for weak convergence of probability measures. Our main result is as follows: Let X be a complete paranormed vector space and Ω an arbitrary metric space, then a sequence {μn} of probability measures is weakly convergent to a probability measure μ if and only if limn→∞∫Ωg(s)dμn=∫Ωg(s)dμ for every bounded continuous function g: Ω → X. A special case is as the following: if X is a Banach space, Ω an arbitrary metric space, then {μn} is weakly convergent to μ if and only if limn→∞∫Ωg(s)dμn=∫Ωg(s)dμ for every bounded continuous function g: Ω → X. Our theorems and corollaries in the article modified or generalized some recent results regarding the convergence of sequences of measures.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

“…There is a rich bibliography concerning the convergence of sequences of measures, besides the above-mentioned studies [1-3]; see, for example, [4][5][6][7][8].…”

Section: Introduction and Terminologymentioning

confidence: 99%

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µ-Integrable Functions and Weak Convergence of Probability Measures in Complete Paranormed Spaces

Zeng

2024

Mathematics

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A comparison among a fuzzy algorithm for image rescaling with other methods of digital image processing

Costarellı,

Sambucını

2024

Constructive Mathematical Analysis

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The aim of this paper is to compare the fuzzy-type algorithm for image rescaling introduced by Jurio et al., 2011, quoted in the list of references, with some other existing algorithms such as the classical bicubic algorithm and the sampling Kantorovich (SK) one. Note that the SK algorithm is a recent tool for image rescaling and enhancement that has been revealed to be useful in several applications to real world problems, while the bicubic algorithm is widely known in the literature. A comparison among the abovementioned algorithms (all implemented in the MatLab programming language) was performed in terms of suitable similarity indices such as the Peak-Signal-to-Noise-Ratio (PSNR) and the likelihood index $S$.

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Vitali Theorems for Varying Measures

Marraffa,

Sambucini

2024

Symmetry

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The classical Vitali theorem states that, under suitable assumptions, the limit of a sequence of integrals is equal to the integral of the limit functions. Here, we consider a Vitali-type theorem of the following form ∫fndmn→∫fdm for a sequence of pair (fn,mn)n and we study its asymptotic properties. The results are presented for scalar, vector and multivalued sequences of mn-integrable functions fn. The convergences obtained, in the vector and multivalued settings, are in the weak or in the strong sense for Pettis and McShane integrability. A list of known results on this topic is cited and new results are obtained when the ambient space Ω is not compact.

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Convergence for varying measures in the topological case

Abstract: In this paper convergence theorems for sequences of scalar, vector and multivalued Pettis integrable functions on a topological measure space are proved for varying measures vaguely convergent.

Cited by 3 publications

References 23 publications

µ-Integrable Functions and Weak Convergence of Probability Measures in Complete Paranormed Spaces

µ-Integrable Functions and Weak Convergence of Probability Measures in Complete Paranormed Spaces

A comparison among a fuzzy algorithm for image rescaling with other methods of digital image processing

Vitali Theorems for Varying Measures

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