In the Mizar system (, ), Józef Białas has already given the one-dimensional Lebesgue measure . However, the measure introduced by Białas limited the outer measure to a field with finite additivity. So, although it satisfies the nature of the measure, it cannot specify the length of measurable sets and also it cannot determine what kind of set is a measurable set. From the above, the authors first determined the length of the interval by the outer measure. Specifically, we used the compactness of the real space. Next, we constructed the pre-measure by limiting the outer measure to a semialgebra of intervals. Furthermore, by repeating the extension of the previous measure, we reconstructed the one-dimensional Lebesgue measure , .
Motivated by the nice characterization of copulas A for which d∞(A, At) is maximal as established independently by Nelsen  and Klement & Mesiar , we study maximum asymmetry with respect to the conditioning-based metric D1 going back to Trutschnig . Despite the fact that D1(A, At) is generally not straightforward to calculate, it is possible to provide both, a characterization and a handy representation of all copulas A maximizing D1(A, At). This representation is then used to prove the existence of copulas with full support maximizing D1(A, At). A comparison of D1- and d∞-asymmetry including some surprising examples rounds off the paper.
In our previous article , we showed complete additivity as a condition for extension of a measure. However, this condition premised the existence of a σ-field and the measure on it. In general, the existence of the measure on σ-field is not obvious. On the other hand, the proof of existence of a measure on a semialgebra is easier than in the case of a σ-field. Therefore, in this article we define a measure (pre-measure) on a semialgebra and extend it to a measure on a σ-field. Furthermore, we give a σ-measure as an extension of the measure on a σ-field. We follow , , and .
Expectation is the fundamental concept in statistics and probability. As two generalizations of expectation, Choquet and Choquet-like expectations are commonly used tools in generalized probability theory. This paper considers the Stolarsky inequality for two classes of Choquet-like integrals. The first class generalizes the Choquet expectation and the second class is an extension of the Sugeno integral. Moreover, a new Minkowski’s inequality without the comonotonicity condition for two classes of Choquet-like integrals is introduced. Our results significantly generalize the previous results in this field. Some examples are given to illustrate the results.
In this paper we study absolutely continuous and σ-finite variational measures corresponding to Mawhin, F- and BV -integrals. We obtain characterization of these σ-finite variational measures similar to those obtained in the case of standard variational measures. We also give a new proof of the Radon-Nikodým theorem for these measures.
Let be the power set of and let be a set function. In this paper, the authors introduce a class of generalized Hausdorff capacities with respect to φ. Some basic properties of including the strong subadditivity are obtained. An equivalent variant of defined via dyadic cubes is also introduced and proved to be Choquet capacity. The authors then prove the boundedness of some maximal operators, such as the Hardy–Littlewood maximal operator, on Lebesgue spaces with respect to . As an application, the predual spaces of weighted Morrey spaces are described via these capacities.
One of the recent advances in the investigation of nonlinear parabolic equations with a measure as forcing term is a paper by F. Petitta in which it has been introduced the notion of renormalized solutions to the initial parabolic problem in divergence form. Here we continue the study of the stability of renormalized solutions to nonlinear parabolic equations with measures but from a different point of view: we investigate the existence and uniqueness of the following nonlinear initial boundary value problems with absorption term and a possibly sign-changing measure data
where Ω is an open bounded subset of ℝN, N ≥ 2, T > 0 and Q is the cylinder (0, T) × Ω, Σ = (0, T) × ∂Ω being its lateral surface, the operator is modeled on the p−Laplacian with , μ is a Radon measure with bounded total variation on Q, b is a C1−increasing function which satisfies 0 < b0 ≤ b′(s) ≤ b1 (for positive constants b0 and b1). We assume that b(u0) is an element of L1(Ω) and h : ℝ ↦ ℝ is a continuous function such that h(s) s ≥ 0 for every |s| ≥ L and L ≥ 0 (odd functions for example). The existence of a renormalized solution is obtained by approximation as a consequence of a stability result. We provide a new proof of this stability result, based on the properties of the truncations of renormalized solutions. The approach, which does not need the strong convergence of the truncations of the solutions in the energy space, turns out to be easier and shorter than the original one.
We derive a new (lower) inequality between Kendall’s τ and Spearman’s ρ for two-dimensional Extreme-Value Copulas, show that this inequality is sharp in each point and conclude that the comonotonic and the product copula are the only Extreme-Value Copulas for which the well-known lower Hutchinson-Lai inequality is sharp.
We prove a Riesz–Herz estimate for the maximal function associated to a capacity C on a metric measure space (X,d,μ). This estimate extends the equivalence for the usual Hardy–Littlewood maximal function Mf and the Riesz–Herz estimate for the capacitary maximal function on ℝn.
Essential tools are the extension of the Wiener–Stein estimate for the distribution function of and the existence of appropriate dyadic cubes in metric measure spaces.
Finally, we obtain the Riesz–Herz estimate for a discrete version of the capacitary maximal function.
The proofs of uniqueness theorems, presented here, allow to extend the earlier results. For example, the following hold: let μ and ν be two finitely additive probabilities on a structure L̃ (for example, L̃ is a pseudo-effect algebra), and let μ be convex-ranged; if there exists an element a ∈ L̃ with 0 < μ(a) < 1 and such that μ(a) = μ(b) ⇒ ν(a) = ν(b) whenever b ∈ L̃, then ν = μ.