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

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Let $X$ be a quasicomplete locally convex Hausdorff space. Let $T$ be a locally compact Hausdorff space and let ${C}_{0}\left(T\right)=\{f\phantom{\rule{0.222222em}{0ex}}T\to I$, $f$ is continuous and vanishes at infinity$\}$ be endowed with the supremum norm. Starting with the Borel extension theorem for $X$-valued $\sigma $-additive Baire measures on $T$, an alternative proof is given to obtain all the characterizations given in [13] for a continuous linear map $u\phantom{\rule{0.222222em}{0ex}}{C}_{0}\left(T\right)\to X$ to be weakly compact.

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.