Structural characteristics of Hewitt–Stromberg outer measures on $$\sigma $$-algebras of metric spaces

In this paper, we introduce a generalized framework for conditional probability in stochastic processes taking values in infinite-dimensional normed spaces. Classical definitions, based on measurability with respect to a conditioning σ-algebra, become inadequate when the available information is restricted to a σ-algebra generated by a finite or countable family of random variables. In such settings, many events of interest are not measurable with respect to the conditioning σ-field, preventing the standard definition of conditional probability. To overcome this limitation, we propose an extension of the coherent conditioning model through the use of Hausdorff measures. The key idea is to exploit the non-equivalence of norms in infinite-dimensional spaces, which gives rise to distinct metric structures and corresponding Hausdorff dimensions for the same events. Conditional probabilities are then defined relative to families of Hausdorff outer measures parameterized by their dimensional exponents. This geometric reformulation allows the notion of conditionality to depend explicitly on the underlying metric and topological properties of the space. The resulting model provides a flexible and coherent framework for analyzing conditioning in infinite-dimensional stochastic systems, with potential implications for Bayesian inference in functional spaces.

The point-to-set principle [J. H. Lutz and N. Lutz, ACM Trans. Comput. Theory, 10(2), 7 (2018).] characterizes the Hausdorff dimension of a subset [Formula: see text] by the effective (or algorithmic) dimension of its individual points. This characterization has been used to prove several results in classical, i.e. without any computability requirements, analysis. Recent work has shown that algorithmic techniques can be fruitfully applied to Marstrand’s projection theorem, a fundamental result in fractal geometry. In this paper, we introduce the notion of optimal oracles for subsets [Formula: see text]. One of the primary motivations of this definition is that, if [Formula: see text] has optimal oracles, then the conclusion of Marstrand’s projection theorem holds for [Formula: see text]. We show that every analytic set has optimal oracles. We also prove that if the Hausdorff and packing dimensions of [Formula: see text] agree, then [Formula: see text] has optimal oracles. Moreover, we show that the existence of sufficiently nice outer measures on [Formula: see text] implies the existence of optimal Hausdorff oracles. In particular, the existence of exact gauge functions for a set [Formula: see text] is sufficient for the existence of optimal Hausdorff oracles, and is therefore sufficient for Marstrand’s theorem. Thus, the existence of optimal oracles extends the currently known sufficient conditions for Marstrand’s theorem to hold. Under certain assumptions, every set has optimal oracles. However, assuming the axiom of choice and the continuum hypothesis, we construct sets which do not have optimal oracles. This construction naturally leads to a generalization of Davies theorem on projections.

The growing amount of small space debris poses a significant threat to spacecrafts with the potential to cause structural damage or even catastrophic failures. Space-based close proximity observation offers an effective alternative to conventional ground-based methods, enabling more precise debris detection through active orbital maneuvers and sensor attitude alignment. However, this approach faces two main challenges. The first is ensuring safe orbit and attitude control to achieve accurate target observation, especially when information about multiple debris is limited; the second is addressing the orbit-attitude coupling problem, which involves a high-dimensional optimization task requiring significant computational resources. This article presents a robust orbit-attitude coupled control method leveraging outer probability measures to address uncertainties caused by incomplete knowledge of multiple debris. To mitigate the computational complexity, a partitioned pattern search algorithm is developed, which incorporates the concept of coordinate descent to find the optimal control solution iteratively. This method decomposes the high-dimensional optimization problem into more manageable subproblems, thereby reducing the computational burden. A convergence analysis is conducted, and an initial guess generation strategy is introduced to further accelerate the optimization process. Simulation results validate its robustness and accuracy in close proximity.

This paper explores coherent upper conditional previsions, a class of nonlinear functionals that generalize expectations while preserving consistency properties. The study focuses on their integral representation using the countably additive Möbius transform, which is possible if coherent upper previsions are defined with respect to a monotone set function of bounded variation. In this work, we prove that an integral representation with respect to a countably additive measure is also possible, on the Borel σ-algebra, even when the coherent upper prevision is defined by the Choquet integral with respect to a Hausdorff measure, which is not of bounded variation. It occurs since Hausdorff outer measures are metric measures, and therefore every Borel set is measurable with respect to them. Furthermore, when the conditioning event has a Hausdorff measure in its own Hausdorff dimension equal to zero or infinity, coherent conditional probability is defined via the countably additive Möbius transform of a monotone set function of bounded variation. The paper demonstrates the continuity of coherent conditional previsions induced by Hausdorff measures.

Against the backdrop of the deep integration of the global digital economy and cross-border e-commerce, and addressing the lack of measure theory and quantitative analysis challenges for high-dimensional dynamic data in this field, this study constructs a measure theory system for cross-border e-commerce markets that combines mathematical rigor and economic interpretability based on Carathéodory’s extension theorem. Based on functional analysis and measure theory, the study defines the market fundamental set as the topological product space of a time index set and a multi-dimensional transaction state space. By constructing a combined structure of a left-open right-closed interval semiring and a power set semiring that satisfies the closure of Boolean algebra operations, an algebraic framework is established for the unified measurement of continuous and discrete variables. On the semiring structure, a σ-finite premeasure integrating Lebesgue measure and counting measure is defined. With the help of the countable covering mechanism generated by outer measure and the measure screening rules of Carathéodory’s measurability condition, the axiomatic extension from premeasure to complete measure on the σ-algebra is completed. Through the verification of Carathéodory’s condition for subsets of null sets and the transmission of outer measure monotonicity, the completeness of the measure space is strictly proved, and the core property that “subsets of null sets must be measurable” is established, providing a solid measure-theoretic foundation for mathematical modeling of cross-border e-commerce markets. At the empirical analysis level, the study uses micro-panel data on global cross-border e-commerce transactions from 2018 to 2024. Through the Kolmogorov-Smirnov test in non-parametric hypothesis testing, the distribution isomorphism between the theoretical measure and empirical data is verified. Based on the measure space theory, a Generalized Method of Moments (GMM) panel regression model is constructed. System GMM and Difference GMM estimation methods are used to handle endogeneity issues. Combined with instrumental variable methods and lag variable techniques, key parameters such as the logarithmic elasticity of economic scale between importing and exporting countries, the spatial decay effect of geographical distance, and the asymmetric inhibitory effect of tariff policies are quantitatively analyzed. A graph neural network model integrating measure theory is innovatively designed. By introducing a completeness regular term, the measure constraints on null sets and their subsets are achieved. Combined with the SHAP value interpretability analysis method, the marginal contribution of each characteristic variable in model decision-making is revealed. The study finds that the constructed measure space not only satisfies the axiomatic requirements of modern measure theory such as completeness and σ-finiteness, but also through the empirical tests of the GMM model and graph neural network, it is confirmed that it can effectively characterize the economic scale effect, spatial distance decay law, and policy sensitivity characteristics in cross-border e-commerce transactions, providing a methodological innovation paradigm based on measure theory for quantitative analysis in the field of international business in the digital economy era.

A major challenge in many multiple space object tracking algorithms lies in the inability to determine a complete probabilistic characterization of the different aspects of the system such as dynamics and observations. In this paper, a robust multiple space object tracking method is developed by leveraging the framework of outer probability measures. The possibility generalized labeled multi-Bernoulli (GLMB) filter is employed to handle epistemic uncertainty in the process of multiple space object tracking, such as the presence of ignorance in dynamical and observation models. In order to initiate the orbital state of birth targets in the context of the possibility GLMB filter, the possibilistic admissible region (PAR+) method is introduced to offer a reliable initial orbit determination based on imperfect information. The developed possibilistic PAR+GLMB scheme provides improved robustness in the case of partial knowledge about the system. The features of the proposed PAR+GLMB method are illustrated using 4 simulated space object tracking case studies.

We investigate the relationship between algorithmic fractal dimensions and the classical local fractal dimensions of outer measures in Euclidean spaces. We introduce global and local optimality conditions for lower semicomputable outer measures. We prove that globally optimal outer measures exist. Our main theorem states that the classical local fractal dimensions of any locally optimal outer measure coincide exactly with the algorithmic fractal dimensions. Our proof uses an especially convenient locally optimal outer measure κ defined in terms of Kolmogorov complexity. We discuss implications for point-to-set principles.

The aim of the study was to obtain phenotypic characteristics of the resource population chickens characterizing the development of the musculoskeletal system and body build for the subsequent study of the structural variability of the chicken genome and the search for gene variants that determine the manifestation of meat productivity phenotypes, including the development of the musculoskeletal system. Materials and methods. The object of the study were chickens of the F0 generation of the Uzbek fighting breed (n = 65) and Cornish (n = 79); hybrids F1 (♂ Cornish x ♀ Uzbek fighting; n = 117) and F2 (♂F1 x ♀ F1; n = 133). At the age of one day, 2 and 4 weeks, the weight (g) of the chickens was determined. Exterior measurements were taken at the age of 4 weeks. The following indicators were estimated (cm): chest girth; metatarsus girth; keel length; metatarsus length. The body conformation indices were calculated. Results. The chickens of the original F0 breeds and F1, F2 hybrids differed significantly in live weight at the age of 2 and 4 weeks. The weight of the Uzbek Fighting breed gene pool chickens was 3,5–4,2 times lower than that of the Cornish breed; the F1 and F2 hybrids had intermediate indicators. In Cornish breed chickens (F0), the coefficient of variability in live weight at the age of 4 weeks was 13,8 %; in Uzbek Fighting breed chickens (F0) – 15,0 %; in F1 and F2 hybrids – 20,5 and 17,6 %, respectively. The maximum values of the massiveness, eirisomy and shortness indices were obtained on Cornish breed chickens, while Uzbek Fighting breed chickens had low indicators; F1 and F2 hybrids are intermediate. Heterosis was established for the relative length of the metatarsus in F1 and F2 hybrids compared to the original breeds. The data obtained proved the feasibility of using the Cornish and Uzbek fighting breeds to create a resource population for the purpose of conducting research to find gene variants that determine the manifestation of meat productivity phenotypes.

Let \mathcal{I} be a surjective closed operator ideal with associated outer measure \gamma_{{\mathcal{I}}} , and let \gamma_{{\mathfrak{I}}} be the extension of \gamma_{{\mathcal{I}}} to bilinear operators. We establish a convexity inequality for the measure \gamma_{{\mathfrak{I}}} of a bilinear operator interpolated by the real method. Our arguments do not require that \mathcal{I} be symmetric.

The measure-preserving, but not necessarily invertible, ergodic transformations of the compact metric space with the Caratheodory measure are considered. The behavior of the Birkhoff sums for integrable and almost everywhere bounded functions with zero mean value in terms of the Caratheodory measure is studied. It is shown that for almost all points of the metric space there is an infinite sequence of "moments of time"; along which the Birkhoff sums tend to zero and at the same moments the trajectory points approach their initial position as close as possible (as in the Poincare return theorem). As an example, we consider the transformation ; of the single segment; closely related to Bernoulli tests.

The concept of soft sets is widely used in reasoning when dealing with vague, incomplete, and imprecise information. Since the introduction of the soft set concept by Molodtsov, the theory of soft sets has become the subject of interest for many researchers who have defined new concepts such as soft measure, soft ?-algebra, and other concepts that are defined in classical measure theory. In the work [5], it was shown that with certain extensions, a soft measure can be constructed on the obtained soft ?-algebra based on soft premeasures on a soft semiring. This study demonstrates that the same conclusion can be reached using a simpler and more natural construction that does not involve the use of a soft exterior measure but rather considers soft sets whose soft measure is zero.

Relevance. Accounting for live weight in reindeer husbandry is an important indicator for breeding, but due to the difficulty of determining it, there is a need to develop a predictive matrix of animal weight through biometric exterior measurements. The purpose of the study is to develop a model for predicting the live weight of reindeer using regression analysis. Methods. The research was carried out using exterior measurements and weighing results of Nenets reindeer (males n = 48, females n = 50) from the Tazovsky district on the Yamal Peninsula at the age of 2 to 9 years. Calculation of statistical parameters, visualization of correlation data and regression analysis using the least squares method were carried out in MS Excel and R-studio.Results. The greatest variability among exterior measurements was noted in the indicators “chest width CW” (9.6%) and “live weight LW” (9.4%). When conducting a correlation analysis, multicollinearity was revealed between the height at the withers HW and the height at the elbow HE r = 0.824 (p ≤ 0.001). High and significant correlations of live weight LW with chest depth CD and chest girth CG r = 0.651 and r = 0.687 (p ≤ 0.001), head length HL r = 0.678 (p ≤ 0.001), height at withers HW r = 0.663 (p ≤ 0.001) and body length BL r = 0.639 (p ≤ 0.001). The most effective model m2 was determined, including chest girth and body length in its structure, the coefficient of determination of which was R2 = 0.70, with multiple R = 0.83, which reflects 70% of the explained variable in the model, with an approximation of 4.2%. As a result, a table was created for predicting the live weight of reindeer using biometric data, which will help simplify selection and breeding work in future populations of hard-to-reach areas.

Context. In general, large solar flares are more efficient at accelerating high-energy electrons than microflares. Nonetheless, sometimes microflares that accelerate electrons to high energies are observed. Their origin is unclear. Aims. We statistically characterized microflares with strikingly hard spectra in the hard X-ray (HXR) range, which means that they are efficient at accelerating high-energy electrons. We refer to these events as “hard microflares”. Methods. We selected 39 hard microflares, based on their spectral hardness estimated from the Solar Orbiter/STIX HXR quicklook light curves in two energy bands. The statistical analysis is built on spectral and imaging information from STIX combined with extreme ultraviolet (EUV) and magnetic field maps from SDO/AIA and SDO/HMI. Results. The key observational result is that all hard microflares in this dataset have one of the footpoint rooted directly within a sunspot (either in the umbra or the penumbra). This clearly indicates that the underlying magnetic flux densities are large. For the events with the classic two-footpoints morphology, the absolute value of the mean line-of-sight magnetic flux density (and vector magnetic field strength) at the footpoint rooted within the sunspot ranges from 600 to 1800 G (1500 to 2500 G), whereas the outer footpoint measures from 10 to 200 G (100 to 400 G), therefore about ten times weaker. In addition, approximately 78% of the hard microflares, which exhibited two HXR footpoints, have similar or even stronger HXR flux from the footpoint rooted within the sunspot. This contradicts the magnetic mirroring scenario. The median footpoint separation, measured through HXR observations, is approximately 24 Mm, which aligns with regular events of similar GOES classes. In addition, about 74% of the events could be approximated by a single-loop geometry, demonstrating that hard microflares typically have a relatively simple morphology. Out of these events, around 54% exhibit a relatively flat flare loop geometry. Conclusions. We conclude that all hard microflares are rooted in sunspots, which implies that the magnetic field strength plays a key role in efficiently accelerating high-energy electrons, with hard HXR spectra associated with strong fields. This key result will allow us to further constrain our understanding of the electron acceleration mechanisms in flares and space plasmas.

Multi-sensor possibility PHD filter for space situational awareness

Determining coefficients for a fractional p-Laplace equation from exterior measurements

The growth abilities of goat kids from the Bulgarian White dairy (BWD) breed in the period up to the end of weaning (90 days) were studied. The study was conducted in the goat farm of the RIMSA - Troyan, Bulgaria, and a total of 44 clinically healthy goat kids of the BWD breed were examined. Some performance characteristics (birth weight, weaning weight, and average daily gain) as well as exterior measurements were measured. The average daily gain for the first, second, and third month for males is (0.154; 0.236; and 0.265 kg), and for females (0.150; 0.166; and 0.255 kg), respectively. At the age of 3 months, twins reliably outperformed singles and triplets in both weight and size, and their average daily gain for the period was 0.211 kg. At birth and weaning, males outperformed females in all exterior indicators. Live weight, growth intensity, and weight development of young animals are factors of primary importance for selection, as they are related to productivity indicators and influence milk yield, fattening capacity, and slaughtering indicators.

Within the Memphis region of Egypt, the Old Kingdom period (approximately 2686–2181 BCE) saw the construction of over 70 pyramids. The extant survey data for these pyramids, derived mainly from architectural studies, have not been substantially updated. The information from this period predominantly encompasses descriptive accounts and schematic representations, focusing on the dimensions, internal chambers, and passageways of the pyramids. Notably absent are detailed documentations that provide insights into the individual stone blocks and the intricacies of their masonry. Consequently, the current understanding of the pyramid construction methodologies of this era remains largely hypothetical. Since 2005, I have been actively engaged in an interdisciplinary 3D survey initiative, focusing on acquiring laser- scanned 3D data of Memphite pyramids. This endeavor aims to analyze their structural forms to elucidate the progression in construction techniques during this epoch. Beginning in 2013, the “Giza 3D Survey” project was initiated as a collaboration between academia and industry. This project sought to comprehensively update the exterior measurement data of the Giza pyramid complex, employing “Structure from Motion/Multi-View Stereo (SfM/MVS)” technologies to generate 3D representations from photographic imagery. This comprehensive approach seeks to unravel the construction methodologies of these ancient structures. As a result of a decade-long survey, it has been discovered that the pyramids of Giza are not merely accumulations of limestone blocks. Instead, they contain a central “core structure” in a step-like formation, a framework made up of “backing stones,” and areas filled with debris and sand, referred to as “chamber method.”

Coherent upper conditional probabilities defined by Hausdorff measures on metric spaces are proven to represent merging opinions with increasing information when the metrics are bi-Lipschitz equivalent .

A σ – finite measure is obtained as the unique common extension of two σ – finite measures defined on two disjoint rings. This is done by first considering an outer measure on a σ – ring containing both disjoint rings. This outer measure then gives rise to the unique extension in question.