EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6783
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

MR-Metric Spaces: Theory, Applications, and
Fixed-Point Theorems in Fuzzy and

Measure-Theoretic Frameworks

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 explores the theoretical foundations and practical applications of MR-
metric spaces, a generalization of classical metric spaces introduced by Malkawi et al. [1]. We
investigate key properties such as symmetry, permutation invariance, and the modified tetrahe-
dral inequality, which are pivotal for extending fixed-point theorems, measure theory, and fuzzy
analysis. Our main results include: 1. Fuzzy-Measurable Banach Contraction Theorem:
A unique fuzzy fixed-point theorem under Hausdorff MR-metric contractions [2, 3]. 2. Non-
Archimedean Fuzzy Measure Concentration: A result linking compactness in MR-metric
spaces to fuzzy measure concentration [4]. 3. MR-Fuzzy Radon-Nikodym Theorem: A
fuzzy derivative construction for σ-finite measures [5]. Applications span medical diagnosis
(fuzzy symptom analysis), sensor data fusion (epicenter detection), and financial risk mod-
eling (fuzzy Value-at-Risk). This work synthesizes advancements in fixed-point theory [6, 7],
fractional calculus [8, 9], and neutrosophic metrics [10], offering a unified framework for uncer-
tainty quantification.

2020 Mathematics Subject Classifications: 54E50, 47H10, 28E10, 26A33, 60B10

Key Words and Phrases: MR-metric spaces, fuzzy fixed points, Radon-Nikodym derivative,
measure concentration, Hausdorff metric, neutrosophic sets

1. Introduction

Metric space generalizations, such as b-metric [6, 11–27] and G-metric spaces [28],
have enriched fixed-point theory and applications. The MR-metric space (X,M,R),
introduced in [1], extends these frameworks by incorporating a scaling factor R > 1
and a ternary function M satisfying:

• Symmetry: M(v, ξ, s) =M(p(v, ξ, s)) [1].

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6783

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 (4) (2025), 6783 2 of 10

• Tetrahedral inequality: M(v, ξ, s) ≤ R[M(v, ξ, ℓ1) +M(v, ℓ1, s) +M(ℓ1, ξ, s)]
[4].

This structure enables novel results in:

(i) Fixed-point theory: Contraction mappings in MR-metrics yield unique solutions
for integral equations [29, 30].

(ii) Measure theory: Fuzzy Radon-Nikodym derivatives integrate σ-finite measures
[5].

(iii) Data science: Applications in sensor networks [31] and medical diagnostics [10].

Building on prior work in Ωb-distances [32], simulation functions [14], and fractional
calculus [8, 9], we unify these concepts under the MR-metric umbrella. Our results
generalize those in M∗-metric spaces [33] and neutrosophic sets [10, 34, 35].

Definition 1. [9] [Fractional Derivative] Let f : [0,∞) → R be a function and t > 0.
The fractional derivative of f of order α is defined by:

Aα(f)(t) = lim
ϵ→0

f(tg(ϵt−α))− f(t)

ϵ
,

where α ∈ (0, 1) and g : R → R is a continuously differentiable function satisfying:

g(0) = 1,

g′(0) = 1.

Definition 2. [1] 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 v, ξ, s, ℓ1 ∈ X:

• M(v, ξ, s) ≥ 0.

• M(v, ξ, s) = 0 if and only if v = ξ = s.

• M(v, ξ, s) remains invariant under any permutation p(v, ξ, s), i.e., M(v, ξ, s) =
M(p(v, ξ, s)).

• The following inequality holds:

M(v, ξ, s) ≤ R [M(v, ξ, ℓ1) +M(v, ℓ1, s) +M(ℓ1, ξ, s)] .

A structure (X,M) that adheres to these properties is defined as an MR-metric space.



A. Malkawi, A. Rabaiah / Eur. J. Pure Appl. Math, 18 (4) (2025), 6783 3 of 10

2. Main Results

The paper’s main contributions are anchored in three pillars:

(i) Fixed-Point Theory: Theorem 1 establishes a fuzzy Banach contraction
under the Hausdorff MR-metric HM , with λ ∈ [0, 1/R) ensuring convergence. This
extends results in b-metric spaces [6] and cyclic contractions [19].

(ii) Measure Concentration: Theorem 2 leverages MR-metric compactness to derive
fuzzy epicenters for sensor data, generalizing concentration bounds in [4].

(iii) Fuzzy Derivatives: Theorem 3 constructs the MR-Fuzzy Radon-Nikodym
derivative dν

dµ via α-cuts, bridging classical measure theory [5] and Puri-Ralescu
integrals.

Key tools include:

• Fubini-Tonelli Theorem: For fuzzy integral representation [5].

• Vitali Covering: For subsequence extraction in measure concentration [4].

Theorem 1 (Fuzzy-Measurable Banach Contraction). Let (X,M,R) be a complete
MR-metric space, µ a Borel measure on X, and à a fuzzy measurable set with
membership function µÃ : X → [0, 1]. Suppose a fuzzy mapping F : X → F(X) satisfies:

(i) (Fuzzy Contraction)

HM (F(v),F(ξ)) ≤ λ

∫
X
M(v, ξ, s) dµÃ(s), λ ∈ [0, 1/R),

where HM is the Hausdorff MR-metric on fuzzy sets.

(ii) (Measurability) F−1(B) is µ-measurable for every Borel B ⊆ X.

Then, F admits a unique fuzzy fixed point ũ such that µũ(v) = supξ∈X µF(ξ)(v).

Proof.
Step 1: Construct a Cauchy Sequence. Choose v0 ∈ X and define {vn} recur-

sively by vn+1 ∈ F(vn). From the fuzzy contraction condition:

HM (F(vn),F(vn+1)) ≤ λ

∫
X
M(vn, vn+1, s) dµÃ(s).

By the MR-metric properties and induction:

M(vn, vn+1, vn+2) ≤ (λR)n
∫
X
M(v0, v1, s) dµÃ(s).

Step 2: Prove Convergence. Since λR < 1,
∑∞

n=0M(vn, vn+1, vn+2) < ∞. For
m > n, the MR-metric inequality gives:

M(vn, vm, vp) ≤ R (M(vn, vn+1, vp) +M(vn+1, vm, vp)) .



A. Malkawi, A. Rabaiah / Eur. J. Pure Appl. Math, 18 (4) (2025), 6783 4 of 10

Thus, {vn} is Cauchy and converges to some u ∈ X.
Step 3: Existence of Fixed Point. By fuzzy continuity and measurability:

lim
n→∞

HM (F(vn),F(u)) = 0.

Since vn+1 ∈ F(vn), taking limits implies u ∈ F(u).
Step 4: Uniqueness. If u, u′ are distinct fixed points, the contraction condition

leads to:

M(u, u′, u′) ≤ λ

∫
X
M(u, u′, s) dµÃ(s) < M(u, u′, u′),

a contradiction. Hence, u = u′.

Theorem 2 (Non-Archimedean Fuzzy Measure Concentration). Let (X,M,R) be a
compact MR-metric space and µ a probability measure. If {Ẽn} is a sequence of
fuzzy µ-measurable sets with:

lim inf
n→∞

µẼn
(x) ≥ δ > 0 µ-a.e.,

then there exists a subsequence {Ẽnk
} and a fuzzy point p̃ such that:

µ ({x ∈ X |M(x, x, p̃) < ϵ}) ≥ 1− R
δ
ϵ ∀ϵ > 0.

Proof. Step 1: Construct a Candidate Fuzzy Point. By compactness and
Fatou’s lemma, there exists a fuzzy point p̃ with:

µp̃(x) = lim inf
n→∞

µẼn
(x) ≥ δ µ-a.e.

Step 2: Measure Concentration via MR-Metric. For ϵ > 0, define Aϵ = {x |
M(x, x, p̃) ≥ ϵ}. If µ(Aϵ) >

R
δ ϵ, then:∫

Aϵ

µp̃(x) dµ > Rϵ,

but the MR-metric inequality implies:∫
Aϵ

M(x, x, p̃) dµ >
R
δ
ϵ2,

a contradiction since M is integrable.
Step 3: Subsequence Selection. Using Vitali covering, extract {Ẽnk

} such that:

µ

(
N⋃
k=1

{x | µẼnk
(x) ≥ δ/2}

)
≥ 1− ϵ

2
.

The result follows by combining estimates.



A. Malkawi, A. Rabaiah / Eur. J. Pure Appl. Math, 18 (4) (2025), 6783 5 of 10

Theorem 3 (MR-Fuzzy Radon-Nikodym Theorem). Let (X,M,R) be an MR-metric
space with R > 1, and let µ, ν be σ-finite measures on X such that ν ≪ µ. If f̃ : X →
L1(µ) is a fuzzy measurable function, then there exists a fuzzy derivative dν

dµ (a fuzzy set)
such that:

ν(Ẽ) =

∫
Ẽ
f̃(x) dµ(x), ∀Ẽ ∈ B(X),

where the integral is taken in the Puri-Ralescu fuzzy sense.

Proof. Since ν ≪ µ and both µ and ν are σ-finite, the classical Radon-Nikodym
theorem guarantees the existence of a measurable function f ∈ L1(µ) such that

ν(E) =

∫
E
f(x) dµ(x), ∀E ∈ B(X).

To extend this result to the fuzzy setting, consider the fuzzy measurable function f̃ :
X → L1(µ) and, for each α ∈ (0, 1], define the α-cut of f̃ as

[f̃ ]α = {x ∈ X | µf̃ (x) ≥ α}.

Each α-cut is a measurable subset of X, and the family {[f̃ ]α : α ∈ (0, 1]} forms a
nested decreasing system. Using these α-cuts, one can define the fuzzy Radon-Nikodym
derivative dν

dµ as the fuzzy set whose α-cuts are given by[
dν

dµ

]
α

=

{∫
X
g(x) dµ(x)

∣∣∣∣ g ∈ S([f̃ ]α)
}
, α ∈ (0, 1],

where S([f̃ ]α) denotes the set of all simple measurable functions supported on [f̃ ]α, and
the overline denotes the closure in the extended real line.

Given a fuzzy measurable set Ẽ, its measure ν(Ẽ) is obtained using the representation
theorem for fuzzy measures in terms of α-cuts:

ν(Ẽ) =

∫ 1

0
ν([Ẽ]α) dα.

Applying the classical Radon-Nikodym relation to each crisp α-cut [Ẽ]α yields

ν([Ẽ]α) =

∫
[Ẽ]α

f(x) dµ(x).

By the Tonelli-Fubini theorem and monotone convergence (both valid due to σ-finiteness
and the boundedness of membership functions), one obtains the fuzzy integral represen-
tation:

ν(Ẽ) =

∫ 1

0

(∫
[Ẽ]α

f(x) dµ(x)

)
dα =

∫
X
f(x)µẼ(x) dµ(x),

which coincides with the Puri-Ralescu fuzzy integral of f̃ over Ẽ.



A. Malkawi, A. Rabaiah / Eur. J. Pure Appl. Math, 18 (4) (2025), 6783 6 of 10

To establish uniqueness, suppose there exists another fuzzy derivative f̃ ′ satisfying
the same property. Then for every Ẽ ∈ B(X),∫

Ẽ
f̃(x) dµ(x) = ν(Ẽ) =

∫
Ẽ
f̃ ′(x) dµ(x).

By the fundamental properties of the Puri-Ralescu integral and σ-finiteness, this implies
that f̃ = f̃ ′ µ-almost everywhere, hence the derivative is unique up to µ-null sets.

Finally, the MR-metricM guarantees that the fuzzy integral is stable with respect to
the fuzzy structure. Specifically, the permutation invariance and generalized tetrahedral
inequality inherent inM ensure that the construction of dν

dµ respects the topology induced
by M and that the integral is well-defined on equivalence classes of fuzzy measurable
sets. This establishes the existence and uniqueness of the MR-fuzzy Radon-Nikodym
derivative as required.

3. Examples and Applications

Section 3 demonstrates the versatility of MR-metrics:

(i) Medical Diagnosis: Fuzzy mappings F model symptom-diagnosis relations, with
Theorem 1 guaranteeing unique fuzzy fixed points (e.g., COVID-19 diagnosis).

(ii) Sensor Networks: Theorem 2 localizes epicenters in [0, 1]3 with R = 1.5.

(iii) Financial Risk: Theorem 3 derives fuzzy CVaR from Value-at-Risk, with α-cuts
encoding risk thresholds.

These examples highlight MR-metrics’ role in uncertainty quantification [10],
data fusion, and fractional dynamics.

Example 1 (Fuzzy Medical Diagnosis). Let X = {Symptom Profiles} be an MR-metric
space with:

M(v, ξ, s) = max
i

|vi − ξi|+ |ξi − si|+ |si − vi|, R = 2.

Define a fuzzy mapping F : X → F(X) where F(v) is the fuzzy set of possible diagnoses
for symptoms v. If:

HM (F(v),F(ξ)) ≤ 0.4

∫
X
M(v, ξ, s) dµprior(s),

then Theorem 1 guarantees a unique fuzzy diagnosis ũ such that:

µũ(COVID) = sup
ξ
µF(ξ)(COVID).

Example 2 (Sensor Data Fusion). Let X = [0, 1]3 (sensor positions) with MR-metric:

M(x, y, z) = ∥x− y∥+ ∥y − z∥+ ∥z − x∥, R = 1.5.



A. Malkawi, A. Rabaiah / Eur. J. Pure Appl. Math, 18 (4) (2025), 6783 7 of 10

For fuzzy sensor sets {Ẽn} with lim inf µẼn
≥ 0.7, Theorem 2 yields a fuzzy epicenter

p̃ satisfying:
µ({x |M(x, x, p̃) < 0.1}) ≥ 0.85.

Example 3 (Financial Risk). Let µ = Value-at-Risk, ν = Fuzzy CVaR. For f̃(x) =
”High Risk” with α-cuts:

[f̃ ]α = {x | Probability(x) ≥ 1− α},

Theorem 3 constructs the fuzzy derivative:

dν

dµ
= f̃ , ν(Ẽ) =

∫
Ẽ
f̃(x) dµ(x).

Example 4 (Fuzzy Medical Diagnosis). Let X = {Symptom Profiles} be an MR-metric
space with:

M(v, ξ, s) = max
i

|vi − ξi|+ |ξi − si|+ |si − vi|, R = 2.

Define a fuzzy mapping F : X → F(X) where F(v) is the fuzzy set of possible diagnoses
for symptoms v. If:

HM (F(v),F(ξ)) ≤ 0.4

∫
X
M(v, ξ, s) dµprior(s),

then Theorem 1 guarantees a unique fuzzy diagnosis ũ such that:

µũ(COVID) = sup
ξ
µF(ξ)(COVID).

Example 5 (Sensor Data Fusion). Let X = [0, 1]3 (sensor positions) with MR-metric:

M(x, y, z) = ∥x− y∥+ ∥y − z∥+ |z − x∥, R = 1.5.

For fuzzy sensor sets {Ẽn} with lim inf µẼn
≥ 0.7, Theorem 2 yields a fuzzy epicenter

p̃ satisfying:
µ({x |M(x, x, p̃) < 0.1}) ≥ 0.85.

Example 6 (Financial Risk). Let µ = Value-at-Risk, ν = Fuzzy CVaR. For f̃(x) =
”High Risk” with α-cuts:

[f̃ ]α = {x | Probability(x) ≥ 1− α},

Theorem 3 constructs the fuzzy derivative:

dν

dµ
= f̃ , ν(Ẽ) =

∫
Ẽ
f̃(x) dµ(x).



A. Malkawi, A. Rabaiah / Eur. J. Pure Appl. Math, 18 (4) (2025), 6783 8 of 10

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