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Dear Colleagues,

Measure theory has long been a cornerstone of modern mathematics, providing a rigorous framework for understanding integration, probability, and geometric structures. Its foundational concepts—such as σ-algebras, measurable functions, and Lebesgue integration—have become indispensable tools in diverse fields, ranging from stochastic analysis and dynamical systems to machine learning and financial mathematics. The interplay between measure theory and related areas—such as geometric measure theory, ergodic theory, non-additive measure and integral theory, game theory, and functional analysis—continues to yield profound insights and innovative applications. In this Special Issue, we seek to showcase cutting-edge research that advances the theory, methods, and interdisciplinary applications of measure theory. We aim to bridge theoretical advancements with practical innovations, fostering dialog between pure mathematicians and applied researchers. By highlighting the synergy between measure theory and its related disciplines, we seek to address open challenges, such as the interplay between Borel and Lebesgue measurability (open problems in Banach spaces), and to explore emerging domains like measure-theoretic machine learning and geometric analysis. We invite contributions that push the boundaries of measure theory, whether with novel proofs, computational breakthroughs, or transformative applications, across various fields of science and engineering.

Prof. Dr. Radko Mesiar
Prof. Dr. Endre Pap
Dr. Anna Rita Sambucini
Guest Editors

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Research

14 pages, 334 KB

Open AccessArticle

Splitting of Conditional Expectations and Liftings in Product Spaces II

by Kazimierz Musiał

Axioms 2026, 15(3), 157; https://doi.org/10.3390/axioms15030157 - 24 Feb 2026

Viewed by 360

Abstract

Let

(X, A, P)

and

(Y, B, Q)

be two probability spaces, R be their skew product on the product

σ

-algebra

A \otimes B

and [...] Read more.

Let

(X, A, P)

and

(Y, B, Q)

be two probability spaces, R be their skew product on the product

σ

-algebra

A \otimes B

and

{(A_{y}, S_{y}) : y \in Y}

be a Q-disintegration of R. Then, let

A ⋇ B

be the

σ

-algebra generated

A \otimes B

and by the family

M : = {E \subset X \times Y : \exists N \in B_{0} \forall y \notin N \hat{S_{y}} (E^{y}) = 0}

and

R_{⋇}

be the extension of R such that

M

becomes the family of

R_{*}

-zero sets (

\hat{S_{y}}

is the completion of

S_{y}

and

B_{0} = {B \in B : Q (B) = 0}

). We prove that there exists a lifting

π

L^{\infty} (R_{⋇})

and liftings

σ_{y}

L^{\infty} (\hat{S_{y}})

y \in Y

, such that sections of

π

determined by Y are lifting invariant (in particular, the sections are measurable), i.e.,

{[π (f)]}^{y} = σ_{y} ({[π (f)]}^{y}) for every y \in Y and every f \in L^{\infty} (R_{⋇}) .

In general, if

π

is an arbitrary lifting on the product, then some sections of

π (f)

may be even nonmeasurable. The main novelty of my paper lies in expanding the domain of the measure in the product to

A ⋇ B

and constructing on such a much larger abstract space the suitable lifting. Such expansions used to be made only in case of topological spaces, where product of marginal Borel sets was replaced by the Borel subsets of the product space. However, several topological technics are then applied, not approachable in the abstract case. The main theorem is a generalization of earlier lifting results, where either separability of

A

in the Frechet–Nikodým pseudometric was assumed or

R ≪ P \times Q

. In case of a separable P and in the case when

R ≪ P \times Q

, a characterization of stochastic processes possessing an equivalent measurable version is presented. The theorem is a strong generalization of earlier results (see the introduction) where it was proved only that the lifting modification of a measurable stochastic process (via the lifting constructed there) is again measurable. Full article

(This article belongs to the Special Issue Measure Theory and Related Topics)

19 pages, 360 KB

Open AccessFeature PaperArticle

Some Reverse Inequalities for Scalar Birkhoff Weak Integrable Functions

by Anca Croitoru, Alina Iosif, Anna Rita Sambucini and Luca Zampogni

Axioms 2026, 15(2), 133; https://doi.org/10.3390/axioms15020133 - 12 Feb 2026

Viewed by 471

Abstract

The Minkowski and Hölder inequalities play an important role in many areas of pure and applied mathematics, such as Convex Analysis, Probabilities, Control Theory, Fixed Point theorems, and Mathematical Economics. Also, non-additive measures, non-additive integrals and set-valued integrals are useful tools in several areas of theoretical and applied mathematics. In this paper we present and prove some Hölder and Minkowski inequality (or reverse inequality) types obtained for Birkhoff weak integrable functions with respect to a non-additive measure. Then, we apply these results to the interval-valued case. Full article

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