### $(L,M)$-fuzzy $\sigma $-algebras.

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A metric space (X,d) is monotone if there is a linear order < on X and a constant c such that d(x,y) ≤ cd(x,z) for all x < y < z in X, and σ-monotone if it is a countable union of monotone subspaces. A planar set homeomorphic to the Cantor set that is not σ-monotone is constructed and investigated. It follows that there is a metric on a Cantor set that is not σ-monotone. This answers a question raised by the second author.

Ψ-density point of a Lebesgue measurable set was introduced by Taylor in [Taylor S.J., On strengthening the Lebesgue Density Theorem, Fund. Math., 1958, 46, 305–315] and [Taylor S.J., An alternative form of Egoroff’s theorem, Fund. Math., 1960, 48, 169–174] as an answer to a problem posed by Ulam. We present a category analogue of the notion and of the Ψ-density topology. We define a category analogue of the Ψ-density point of the set A at a point x as the Ψ-density point at x of the regular open...

We show how to capture the gradient concentration of the solutions of Dirichlet-type problems subjected to large sources of order $\frac{1}{\sqrt{\epsilon}}$ concentrated on an $\epsilon $-neighborhood of a hypersurface of the domain. To this end we define the gradient Young-concentration measures generated by sequences of finite energy and establish a very simple characterization of these measures.

We show how to capture the gradient concentration of the solutions of Dirichlet-type problems subjected to large sources of order $\frac{1}{\sqrt{\epsilon}}$ concentrated on an ε-neighborhood of a hypersurface of the domain. To this end we define the gradient Young-concentration measures generated by sequences of finite energy and establish a very simple characterization of these measures.

Consider an experiment with d+1 possible outcomes, d of which occur with probabilities $x\u2081,...,{x}_{d}$. If we consider a large number of independent occurrences of this experiment, the probability of any event in the resulting space is a polynomial in $x\u2081,...,{x}_{d}$. We characterize those polynomials which arise as the probability of such an event. We use this to characterize those x⃗ for which the measure resulting from an infinite sequence of such trials is good in the sense of Akin.

En este artículo revisamos un famoso teorema, descubierto por H. Steinhaus en 1936, en el que se da una condición suficiente que permite obtener las funciones coordenadas de una curva que llena el cuadrado unidad. Ponemos de manifiesto que el recíproco de este teorema no se cumple para la curva de Lebesgue. Aquí proponemos un teorema de caracterización de curvas que llenan el espacio, basado en una condición de llenado. Asimismo, damos una caracterización constructiva de esta condición de llenado...

We give a complete characterization of tribes with respect to the Łukasiewicz $t$-norm, i. e., of systems of fuzzy sets which are closed with respect to the complement of fuzzy sets and with respect to countably many applications of the Łukasiewicz $t$-norm. We also characterize all operations with respect to which all such tribes are closed. This generalizes the characterizations obtained so far for other fundamental $t$-norms, e. g., for the product $t$-norm.

Is the Lebesgue measure on [0,1]² a unique product measure on [0,1]² which is transformed again into a product measure on [0,1]² by the mapping ψ(x,y) = (x,(x+y)mod 1))? Here a somewhat stronger version of this problem in a probabilistic framework is answered. It is shown that for independent and identically distributed random variables X and Y constancy of the conditional expectations of X+Y-I(X+Y > 1) and its square given X identifies uniform distribution either absolutely continuous or discrete....