EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6528 ISSN 1307-5543 – ejpam.com Published by New York Business Global Applications of MR-Metric Spaces in Measure Theory and Convergence Analysis Abed Al-Rahman M. Malkawi1,∗, Ayat M. Rabaiah1 1 Department of Mathematics, Faculty of Arts and Science, Amman Arab University, Amman 11953, Jordan Abstract. This paper investigates the role of MR-metric spaces in the fields of measure theory and convergence analysis. We analyze how measures defined through a triadic metric function reveal key characteristics such as σ-finiteness and absolute continuity. Furthermore, we ex- plore the convergence behavior of sequences within the MR-metric framework, highlighting their relevance in areas like stochastic processes, optimization, and data science. The findings offer valu- able perspectives on probability distributions that incorporate triadic dependencies, the conditions necessary for stability in machine learning, and the clustering behavior within complex networks. These discoveries pave the way for extending traditional metric space theory into more advanced settings, including those involving multi-agent systems and high-dimensional analyses. 2020 Mathematics Subject Classifications: 54E50, 28A12, 60B10, 46E27, 37A50 Key Words and Phrases: MR−metric space, Absolute continuity, Measure theory, σ-finiteness, Fixed point theorems, MR-convergent, Optimization 1. Introduction This paper presents an in-depth exploration of MR-metric spaces, particularly their application within the contexts of measure theory and convergence analysis. MR-metric spaces, characterized by a triadic metric function, provide a novel approach to understand- ing complex relationships in various mathematical domains. By leveraging this triadic structure, we examine foundational properties such as σ-finiteness and absolute continu- ity, which are crucial for the rigorous analysis of measures. These properties enable a more robust framework for investigating diverse mathematical phenomena, including the behavior of stochastic processes. Furthermore, the study explores how these metric spaces contribute to the stability of machine learning models, offering new insights into optimization problems and enhancing ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6528 Email addresses: a.malkawi@aau.edu.jo and math.malkawi@gmail.com (A. Malkawi), a.rabaieha@aau.edu.jo (A. Rabaiah) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. Malkawi, A. Rabaiah / Eur. J. Pure Appl. Math, 18 (3) (2025), 6528 2 of 12 algorithmic performance in the broader field of data science. By addressing the intri- cate connections between MR-metric spaces and probability distributions that incorpo- rate triadic dependencies, this research provides a deeper understanding of the underlying mathematical principles. The results also open up promising new avenues for extending the theory of metric spaces, particularly in high-dimensional settings and complex sys- tems involving multiple interacting agents, such as those found in network analysis and multi-agent models. For further details, we refer readers to the works cited in [1–21]. Definition 1. [22] Consider a non-empty set X ̸= ∅ and a real number R > 1. A function M : X × X × X → [0,∞) is termed an MR-metric if it satisfies the following conditions for all υ, ξ,ℑ ∈ X: • (M1) M(υ, ξ,ℑ) ≥ 0. • (M2) M(υ, ξ,ℑ) = 0 if and only if υ = ξ = ℑ. • (M3)M(υ, ξ,ℑ) remains invariant under any permutation p(υ, ξ,ℑ), i.e.,M(υ, ξ,ℑ) = M(p(υ, ξ,ℑ)). • (M4) The following inequality holds: M(υ, ξ,ℑ) ≤ R [M(υ, ξ, ℓ1) +M(υ, ℓ1,ℑ) +M(ℓ1, ξ,ℑ)] . A structure (X,M) that adheres to these properties is defined as an MR-metric space. Definition 2. [22] Consider a sequence {υin} in an MR-metric space (X,M). This sequence is said to be MR-convergent if there exists an element υi1 ∈ X such that for any ϵ > 0, there exists a positive integer N satisfying the condition M(υin, υim, υi1) < ϵ, for all m,n ≥ N. In this case, we say that {υin} converges in the MR-metric sense to υi1, and we refer to υi1 as the limit of the sequence. Definition 3. [22] A sequence {υin} in an MR-metric space (X,M) is termed MR- Cauchy if for every ϵ > 0, there exists a positive integer N such that the inequality M(υin, υim, υip) < ϵ holds for all m,n, p ≥ N. Definition 4. [22] An MR-metric space (X,M) is said to be bounded if there exists a constant L > 0 such that M(υ, ξ,ℑ) ≤ L for all υ, ξ,ℑ ∈ X. In this case, the function M is called an MR-bound for the metric. Definition 5 ([23], Definition 1.1). A measure space is a triplet (X,Σ, µ), where: A. Malkawi, A. Rabaiah / Eur. J. Pure Appl. Math, 18 (3) (2025), 6528 3 of 12 • X is a non-empty set. • Σ is a σ-algebra on X, which satisfies the following properties: (i) X ∈ Σ. (ii) If A ∈ Σ, then the complement Ac ∈ Σ. (iii) If {An}∞n=1 ⊆ Σ, then ⋃∞ n=1An ∈ Σ. • µ : Σ → [0,∞] is a function satisfying: (i) Non-negativity: µ(A) ≥ 0 for all A ∈ Σ. (ii) Null Empty Set: µ(∅) = 0. (iii) Countable Additivity (σ-additivity): If {An}∞n=1 is a countable collection of disjoint sets in Σ, then: µ ( ∞⋃ n=1 An ) = ∞∑ n=1 µ(An). The function µ is called a measure, and X is referred to as the measurable space. Definition 6 ([24], Section 2). A measure µ is said to be σ-finite if there exists a countable collection of measurable sets {Xn}∞n=1 ⊆ Σ such that: X = ∞⋃ n=1 Xn and µ(Xn) <∞ for every n. Definition 7 ([25], Section 3.1). A measure µ is called absolutely continuous with respect to another measure ν (denoted µ≪ ν) if for every measurable set A ∈ Σ, ν(A) = 0 ⇒ µ(A) = 0. 2. Main Result Theorem 1. Let (X,M) be an MR-metric space, and let µ be a measure on X such that: µ(A) = ∫ A M(υ, ξ,ℑ) dµ(υ) dµ(ξ) dµ(ℑ), (1) for measurable sets A ⊆ X. If M satisfies (M4) with a finite measure space, then µ is σ-finite and absolutely continuous. Proof. Step 1: Proving σ-finiteness To demonstrate that µ is σ-finite, we need to decompose the space X into a countable collection of subsets {Xn} such that µ(Xn) <∞ for all n. A. Malkawi, A. Rabaiah / Eur. J. Pure Appl. Math, 18 (3) (2025), 6528 4 of 12 First, since M satisfies (M4), we have the following inequality: M(υ, ξ,ℑ) ≤ R [M(υ, ξ, ℓ1) +M(υ, ℓ1,ℑ) +M(ℓ1, ξ,ℑ)] , (2) where R is a constant greater than 1. This inequality provides a useful upper bound on the MR-metric. Next, for any measurable set A ⊆ X, we integrate both sides of the inequality:∫ A M(υ, ξ,ℑ) dµ(υ) dµ(ξ) dµ(ℑ) ≤ R [ ∫ A M(υ, ξ, ℓ1) dµ(υ) dµ(ξ) dµ(ℑ) + ∫ A M(υ, ℓ1,ℑ) dµ(υ) dµ(ξ) dµ(ℑ) + ∫ A M(ℓ1, ξ,ℑ) dµ(υ) dµ(ξ) dµ(ℑ) ] . (3) This shows that µ(A) is controlled by the integrals of M on smaller subsets. We can now use the fact that M is bounded for all sets and the measure µ is finite on these subsets. Thus, it is possible to partition X into a countable union of subsets {Xn} such that µ(Xn) <∞ for all n, proving that µ is σ-finite. Step 2: Proving Absolute Continuity Now, we will prove that µ is absolutely continuous with respect to the measure µ. Suppose that for some measurable set A ⊆ X, we have µ(A) = 0. We aim to show that this implies that M(υ, ξ,ℑ) = 0 almost everywhere on A. By the definition of µ, we have: µ(A) = ∫ A M(υ, ξ,ℑ) dµ(υ) dµ(ξ) dµ(ℑ) = 0. (4) Since M is non-negative, we conclude that the integrand must be zero almost everywhere on A. In other words, M(υ, ξ,ℑ) = 0 for µ-almost all (υ, ξ,ℑ) ∈ A. Because M(υ, ξ,ℑ) = 0 almost everywhere, the measure µ of A must be zero, which shows that A is a null set in terms of µ. This proves that µ is absolutely continuous. Conclusion We have shown that µ is both σ-finite and absolutely continuous. Therefore, µ is absolutely continuous with respect to the product measure µ(υ)µ(ξ)µ(ℑ), completing the proof. Example 1. Consider the space X = R with the MR-metric defined as: M(υ, ξ,ℑ) = |υ − ξ|+ |ξ −ℑ|+ |ℑ − υ|. (5) Define the measure µ on X as the Lebesgue measure. The measure of a set A ⊆ X is given by: µ(A) = ∫ A M(υ, ξ,ℑ) dυ dξ dℑ. (6) We now analyze its properties in detail: 1. σ-finiteness: A. Malkawi, A. Rabaiah / Eur. J. Pure Appl. Math, 18 (3) (2025), 6528 5 of 12 • A measure µ is called σ-finite if there exists a countable collection of measurable sets {Xn} such that: X = ∞⋃ n=1 Xn and µ(Xn) <∞ for all n. (7) • Consider the sequence of bounded intervals Xn = [−n, n] ⊂ R. Clearly, ⋃∞ n=1Xn = R, covering the entire space. • For each Xn, the integral defining µ(Xn) is: µ(Xn) = ∫ Xn ∫ Xn ∫ Xn M(υ, ξ,ℑ) dυ dξ dℑ. (8) • Since M(υ, ξ,ℑ) is finite for bounded sets, this integral converges, implying that µ(Xn) <∞ for each n. • Thus, µ is σ-finite. 2. Absolute continuity: • A measure µ is absolutely continuous with respect to the Lebesgue measure λ if for every measurable set A: λ(A) = 0 =⇒ µ(A) = 0. (9) • Suppose A ⊆ X is a measurable set with Lebesgue measure λ(A) = 0. This means that the set has no volume in the standard Lebesgue sense. • By the definition of µ(A): µ(A) = ∫ A M(υ, ξ,ℑ) dυ dξ dℑ. (10) • Since the Lebesgue measure of A is zero, the triple integral above must also be zero because integration over a null set yields zero measure. • Thus, we conclude that µ(A) = 0, establishing absolute continuity of µ with respect to the Lebesgue measure. Hence, we rigorously confirm that µ is both σ-finite and absolutely continuous. Theorem 2. Let (X,M) be an MR-metric space with a measure µ such that for a sequence {υn} in X, M(υn, υn−1, υn−2) → 0 as n→ ∞. (11) Then, υn converges to some υ∗ ∈ X in measure, meaning that for every ϵ > 0, µ ({υ ∈ X :M(υn, υ ∗, υ∗) ≥ ϵ}) → 0 as n→ ∞. (12) A. Malkawi, A. Rabaiah / Eur. J. Pure Appl. Math, 18 (3) (2025), 6528 6 of 12 Proof. We are given that M(υn, υn−1, υn−2) → 0 as n → ∞, which implies that the sequence {υn} is Cauchy in the MR-metric space (X,M). Step 1: Verifying the Cauchy Property The given conditionM(υn, υn−1, υn−2) → 0 as n→ ∞ implies that for every ϵ > 0, there exists an integer N such that for all n ≥ N , M(υn, υn−1, υn−2) < ϵ 3 . By the MR-metric inequality, M(υn, υm, υk) ≤ R [M(υn, υn−1, υn−2) +M(υn−1, υm, υk) +M(υm, υk, υn)] . Since each term on the right-hand side approaches zero, it follows that M(υn, υm, υk) → 0 as n,m, k → ∞, establishing the Cauchy property. Step 2: Existence of Limit Since (X,M) is a complete MR-metric space, every Cauchy sequence converges to a unique limit υ∗ ∈ X. Thus, there exists υ∗ ∈ X such that lim n→∞ υn = υ∗. This ensures that υn is converging in the MR-metric space. Step 3: Convergence in Measure To prove convergence in measure, we analyze the set An = {υ ∈ X :M(υn, υ ∗, υ∗) ≥ ϵ}. For large n, we haveM(υn, υ ∗, υ∗) < ϵ, meaning An shrinks as n increases. The measure µ is σ-finite, meaning there exist countable subsets of finite measure covering X. By absolute continuity, since M(υn, υ ∗, υ∗) → 0, the measure µ(An) → 0 as well. Conclusion Thus, we conclude that µ ({υ ∈ X :M(υn, υ ∗, υ∗) ≥ ϵ}) → 0 as n→ ∞. This completes the proof. Example 2. Consider the MR-metric space (R,M) where the metric is given by M(υ, ξ,ℑ) = |υ − ξ|+ |ξ −ℑ|+ |ℑ − υ|. (13) Let the sequence υn = 1 n for n ∈ N. We verify that: M(υn, υn−1, υn−2) = ∣∣∣∣ 1n − 1 n− 1 ∣∣∣∣+ ∣∣∣∣ 1 n− 1 − 1 n− 2 ∣∣∣∣+ ∣∣∣∣ 1 n− 2 − 1 n ∣∣∣∣ . As n → ∞, each term tends to zero, so M(υn, υn−1, υn−2) → 0, satisfying the hypothesis of the theorem. The limit of υn is clearly υ∗ = 0, and we check convergence in measure. For any ϵ > 0, define An = {υ ∈ R :M(υn, 0, 0) ≥ ϵ}. For large enough n, M(υn, 0, 0) = 3|υn| < ϵ, so An eventually becomes empty, implying µ(An) → 0. Thus, υn → 0 in measure. A. Malkawi, A. Rabaiah / Eur. J. Pure Appl. Math, 18 (3) (2025), 6528 7 of 12 3. Applications of MR-Metric Space Theorems 3.1. Applications of the First Theorem The first theorem concerns measure theory in MR-metric spaces, establishing that any measure satisfying the given integral condition is σ-finite and absolutely contin- uous, provided that M satisfies (M4) with a finite measure space. Possible applications include: 3.1.1. Measure Theory and Integration The given theorem plays a crucial role in measure theory and integration in MR-metric spaces , where distances between points are determined by a triple metric function M instead of the conventional pairwise metric. Below, we discuss its implications in absolute continuity, finiteness of measures, and probability density functions (PDFs). • Preservation of Absolute Continuity and Finiteness of Measures: – In classical measure theory, a measure µ is absolutely continuous with re- spect to another measure ν if for every measurable set A, ν(A) = 0 implies µ(A) = 0. – The theorem ensures that measures defined using M(υ, ξ,ℑ) satisfy absolute continuity , meaning that regions of measure zero remain negligible under the transformation imposed by M . – Additionally, the theorem establishes that µ is σ-finite , guaranteeing that space X can be decomposed into countable measurable subsets of finite measure, making integration well-defined. • Existence of a Probability Density Function (PDF) in MR-Metric Spaces: – In probability theory, an absolutely continuous probability measure admits a density function f(υ) such that: µ(A) = ∫ A f(υ) dυ. – The theorem ensures the existence of such a function when the probability distribution depends on triadic interactions rather than pairwise distances. – For example, consider a probability distribution over a space X where the like- lihood of a point υ is influenced by two additional reference points ξ and ℑ. The measure may be defined as: µ(A) = ∫ A M(υ, ξ,ℑ) dµ(υ) dµ(ξ) dµ(ℑ), where M(υ, ξ,ℑ) models the joint influence of three points rather than traditional pairwise interactions. A. Malkawi, A. Rabaiah / Eur. J. Pure Appl. Math, 18 (3) (2025), 6528 8 of 12 – A practical application arises in spatial statistics, where the probability of an event depends on the location of three interacting sites (e.g., modeling geological formations, epidemic spread, or network communications). – Another example is in machine learning , where triplet-based distance functions (such as in triplet loss functions) are used to define similarity measures based on the interaction of three feature vectors. 3.1.2. Applications in Dynamical Systems and Ergodic Theory Dynamical systems and ergodic theory study the long-term behavior of systems evolving over time, particularly those governed by measure-preserving transformations . In tra- ditional ergodic theory, measures are often defined in terms of pairwise dependencies . However, in systems where interactions are inherently tripartite, the theorem ensures that any ergodic measure defined via the three-point metric function M(υ, ξ,ℑ) remains finite and absolutely continuous, making it suitable for rigorous probabilistic analysis. • Ergodic Measures and Higher-Order Dependencies: – In classical ergodic theory, a measure µ is called ergodic if every invariant set under the transformation T is either of full or zero measure. – The theorem ensures that when a measure is defined through the three-point metric function M(υ, ξ,ℑ), it retains its ergodic properties while remaining finite and absolutely continuous . – This is particularly important in stochastic systems with memory , where the state of a system at time n depends not just on the previous state but on two or more past states. • Tripartite Markov Processes and Measure Evolution: – Many real-world stochastic systems do not follow a simple first-order Markov property but instead involve higher-order dependencies. – Consider a tripartite Markov process, where the transition probabilities depend on the current state and two prior states: P (Xn+1 | Xn, Xn−1, Xn−2). – The theorem ensures that when such a system is described using an MR-metric space, the induced measure remains well-behaved, allowing for meaningful long-term statistical analysis . – Applications include: ∗ Stock market modeling , where an asset’s price evolution may depend on its value over multiple previous time steps. ∗ Genetic sequence evolution , where the probability of a mutation may be influenced by a combination of prior genetic states. A. Malkawi, A. Rabaiah / Eur. J. Pure Appl. Math, 18 (3) (2025), 6528 9 of 12 ∗ Neural activity models , where the activation of a neuron may depend on the signals from a combination of previous stimuli rather than just the last input. • Applications in Physics and Ergodic Theory: – In statistical mechanics , tripartite interactions arise naturally in models of interacting particles, such as in spin-glass systems , where the behavior of a particle depends on multiple neighboring interactions. – In thermodynamics , the evolution of macrostates in a system with long-range dependencies can be analyzed using a measure induced by M(υ, ξ,ℑ). – In dynamical astronomy , the behavior of a celestial body in an N-body problem (such as a planetary system) often requires a metric that accounts for more than just pairwise gravitational effects. – The theorem ensures that such measure-preserving systems remain statistically predictable and analyzable , supporting long-term stability analyses in ergodic theory. 3.2. Applications of the Second Theorem The second theorem states that if a sequence {υn} in an MR-metric space satisfies M(υn, υn−1, υn−2) → 0 as n→ ∞, then υn converges to some υ∗ ∈ X in measure. Applications include: 3.2.1. Convergence in Probability and Stochastic Analysis • This result provides a generalization of Cauchy sequences in MR-metric spaces, ensuring that under the given condition, sequences exhibit a form of probabilistic convergence. • It can be used to study stochastic processes where transitions depend on three previous states rather than just one (e.g., higher-order Markov chains). 3.2.2. Machine Learning and Optimization • In algorithms that rely on a triplet-based distance function, such as triplet loss in deep learning, this theorem guarantees that optimization processes involving triple relations lead to a well-defined limit. • It can be useful in metric learning applications where distances between three points determine learning trajectories. A. Malkawi, A. Rabaiah / Eur. J. Pure Appl. Math, 18 (3) (2025), 6528 10 of 12 3.2.3. Geometric Analysis and Fixed Point Theory • If a dynamical system is governed by a three-point metric function, this theorem ensures that iterative processes converge in measure, which is useful in studying generalized contraction mappings. • It can be applied in establishing fixed points in non-Euclidean geometries or spaces where standard metric properties do not hold. 3.3. Computational Implications The theoretical framework of MR-metric spaces and their associated measures opens avenues for computational exploration. Below, we highlight potential implications: • Algorithmic Complexity: The triadic nature of the MR-metric M(v, ξ,ℑ) may increase computational overhead compared to pairwise metrics, as evaluating M requires O(n3) operations for n points. However, symmetry (M3) and boundedness (M4) could be exploited to optimize calculations, e.g., by caching repeated terms or pruning negligible contributions. • High-Dimensional Data: In machine learning, MR-metrics could model higher- order dependencies in data (e.g., triplet interactions in graph embeddings). While this enriches representation, scalability challenges arise. Approximation techniques, such as sampling or low-rank tensor decompositions, might mitigate these costs. • Convergence Verification: Theorem 2’s condition M(vn, vn−1, vn−2) → 0 sug- gests iterative algorithms could leverage MR-metrics to monitor convergence in multi-agent systems or gradient-based optimization. Early stopping criteria could adapt this condition numerically. • Speculative Applications: Quantum computing might benefit from MR-metrics’ triadic structure, as quantum states naturally encode multi-particle correlations. 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