Abstract: The purpose of this paper is to discuss the properties of regularity and strong regularity of fuzzy measure on metric spaces following the previous results. Some properties are defined with the help of nulladditivity such as inner\outer regularity and the regularity of fuzzy measure. We define the strong regularity of fuzzy measures and show our result that the null-additive fuzzy mea...

It is known by the Conley’s theorem that the chain recurrent set CR(φ) of a deterministic flow φ on a compact metric space is the complement of the union of sets B(A) − A, where A varies over the collection of attractors and B(A) is the basin of attraction of A. It has recently been shown that a similar decomposition result holds for random dynamical systems on noncompact separable complete met...

We present a proof of a Ramsey-type theorem for infinite metric spaces due to Matoušek. Then we show that for every K > 1 every uncountable Polish space has a perfect subset that K-bi-Lipschitz embeds into the real line. Finally we study decompositions of infinite separable metric spaces into subsets that, for some K > 1, K-bi-Lipschitz embed into the real line.

We shall continue to discuss further properties of null-additive fuzzy measure on metric spaces following the previous results. Under the null-additivity condition, some properties of the inner/outer regularity and the regularity of fuzzy measure are shown. Also the strong regularity of fuzzy measure is discussed on complete separable metric spaces. As an application of strong regularity, we pr...

The purpose of this article is to prove existence of mass minimizing integral currents with prescribed possibly non-compact boundary in all dual Banach spaces and furthermore in certain spaces without linear structure, such as injective metric spaces and Hadamard spaces. We furthermore prove a weak -compactness theorem for integral currents in dual spaces of separable Banach spaces. Our theorem...

We show that the Lipschitz structure of a separable quasi-Banach space does not determine, in general, its linear structure. Using the notion of the Arens-Eells p-space over a metric space for 0 < p ≤ 1 we construct examples of separable quasi-Banach spaces which are Lipschitz isomorphic but not linearly isomorphic.

M. Gromov [7] suggested to use coarse embeddings into a Hilbert space or into a uniformly convex space as a tool for solving some of the well-known problems. G. Yu [21] and G. Kasparov and G. Yu [11] have shown that this is indeed a very powerful tool. On the other hand, there exist separable metric spaces ([6] and [5, Section 6]) which are not coarsely embeddable into Hilbert spaces. In [9] (s...

We introduce a condition, uniform payoff security, for games with separable metric strategy spaces and payoffs bounded and measurable in players’ strategies. We show that if any such metric game G is uniformly payoff secure, then its mixed extension G is payoff secure. We also establish that if a uniformly payoff secure metric game G has compact strategy spaces, and if its mixed extension G has...