We consider the Monge-Kantorovich transport problem in a purely measure theoretic setting, i.e. without imposing continuity assumptions on the cost function. It is known that transport plans which are concentrated on c-monotone sets are optimal, provided the cost function c is either lower semi-continuous and finite, or continuous and may possibly attain the value ∞. We show that this is true i...

In this note, the concepts of strong autocontinuity from below and strong converse autocontinuity from below of set function are introduced. By using four types of autocontinuity from below of monotone measure, the relationship between convergence in measure and pseudo-convergence in measure for sequence of measurable function are discussed.

In the cumulative prospect theory (CPT) the preference (utility functional) is represented by the difference of two Choquet integrals. We investigate representations of the corresponding functional by two fuzzy integrals, specially the difference representations of the asymmetric Choquet integral with respect to a signed fuzzy measure with bounded chain variation. We discuss non-monotone, real ...

We prove that the 1-dimensional Hausdorff measure restricted to a simple real analytic curve γ : R → R , N ≥ 2, is locally 1-monotone.

The theory of weighted inequalities for the Hardy operator, acting on monotone functions in R+, was first introduced in [2]. Extensions of these results to higher dimensions have been considered only in very specific cases: in particular, in the diagonal case, for p = 1 only (see [3]). The main difficulty in this context is that the level sets of the monotone functions are not totally ordered, ...

In this note, we give four versions of Egoroff’s theorem in non-additive measure theory by using the condition (E) and the pseudo-condition (E) of set function and the duality relations between the conditions. These conditions employed are not only sufficient, but also necessary for the four kinds of Egoroff’s theorem, respectively.

Let (L,4) be a finite distributive lattice, and suppose that the functions f1, f2 :L→ R are monotone increasing with respect to the partial order 4. Given μ a probability measure on L, denote by E(fi) the average of fi over L with respect to μ, i = 1,2. Then the FKG inequality provides a condition on the measure μ under which the covariance, Cov(f1, f2) := E(f1f2)−E(f1)E(f2), is nonnegative. In...

The theory of weighted inequalities for the Hardy operator, acting on monotone functions in R+, was first introduced in [2]. Extensions of these results to higher dimension have been considered only in very specific cases. In particular, in the diagonal case, only for p = 1 (see [5]). The main difficulty in this context is that the level sets of the monotone functions are not totally ordered, c...

Let (L, ) be a finite distributive lattice, and suppose that the functions f1, f2 :L→ R are monotone increasing with respect to the partial order . Given μ a probability measure on L, denote by E(fi) the average of fi over L with respect to μ, i = 1,2. Then the FKG inequality provides a condition on the measure μ under which the covariance, Cov(f1, f2) := E(f1f2) − E(f1)E(f2), is nonnegative. I...

<p style='text-indent:20px;'>In this paper, we define a renormalized dissipative measure-valued (rDMV) solution of the compressible magnetohydrodynamics (MHD) equations with non-monotone pressure law. We prove existence rDMV solutions and establish suitable relative energy inequality. And obtain weak (measure-valued)-strong uniqueness property help inequality.</p>