Duality for Outer $$L^p_\mu (\ell ^r)$$ Spaces and Relation to Tent Spaces

We study the outer $$L^p$$ L p spaces introduced by Do and Thiele on sets endowed with a measure and an outer measure. We prove that, in the case of finite sets, for $$1< p \leqslant \infty , 1 \leqslant r < \infty $$ 1 < p ⩽ ∞ , 1 ⩽ r < ∞ or $$p=r \in \{ 1, \infty \}$$ p = r ∈ { 1 , ∞ } , the outer $$L^p_\mu (\ell ^r)$$ L μ p ( ℓ r ) quasi-norms are equivalent to norms up to multiplicative constants uniformly in the cardinality of the set. This is obtained by showing the expected duality properties between the corresponding outer $$L^p_\mu (\ell ^r)$$ L μ p ( ℓ r ) spaces uniformly in the cardinality of the set. Moreover, for $$p=1, 1 < r \leqslant \infty $$ p = 1 , 1 < r ⩽ ∞ , we exhibit a counterexample to the uniformity in the cardinality of the finite set. We also show that in the upper half space setting the desired properties hold true in the full range $$1 \leqslant p,r \leqslant \infty $$ 1 ⩽ p , r ⩽ ∞ . These results are obtained via greedy decompositions of functions in the outer $$L^p_\mu (\ell ^r)$$ L μ p ( ℓ r ) spaces. As a consequence, we establish the equivalence between the classical tent spaces $$T^p_r$$ T r p and the outer $$L^p_\mu (\ell ^r)$$ L μ p ( ℓ r ) spaces in the upper half space. Finally, we give a full classification of weak and strong type estimates for a class of embedding maps to the upper half space with a fractional scale factor for functions on $$\mathbb {R}^d$$ R d .

Coherent lower and upper conditional previsions defined by Hausdorff inner and outer measures to represent the role of conscious and unconscious thought in human decision making

The model of coherent lower and upper conditional previsions, based on Hausdorff inner and outer measures, is proposed to represent the preference orderings and the equivalences, respectively assigned by the conscious and unconscious thought in human decision making under uncertainty. Complexity of partial information is represented by the Hausdorff dimension of the conditioning event. When the events, that describe the decision problem, are measurable is represented to the s-dimensional Hausdorff outer measure, where s is the Hausdorff dimension of the conditioning event, an optimal decision can be reached. The model is applied and discussed in Linda’s Problem and the conjunction fallacy is resolved.

Reconstruction of the One-Dimensional Lebesgue Measure

SummaryIn the Mizar system ([1], [2]), Józef Białas has already given the one-dimensional Lebesgue measure [4]. 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 [7], [3].

On a New Kind of Collection of Subsets Noted by δ–field and Some Concepts Defined on δ–field

The objective of this paper is, first, study a new collection of sets such as field and we discuss the properties of this collection. Second, introduce a new concepts related to the field such as measure on field, outer measure on field and we obtain some important results deals with these concepts. Third, introduce the concept of null-additive on field as a generalization of the concept of measure on field. Furthermore, we establish new concept related to - field noted by weakly null-additive on field as a generalizations of the concepts of measure on and null-additive. Finally, we introduce the restriction of a set function on field and many of its properties and characterizations are given.

Weighted Variable Exponent Sobolev spaces on metric measure spaces

In this article we define the weighted variable exponent-Sobolev spaces on arbitrary metric spaces, with finite diameter and equipped with finite, positive Borel regular outer measure. We employ a Hajlasz definition, which uses a point wise maximal inequality. We prove that these spaces are Banach, that the Poincaré inequality holds and that lipschitz functions are dense. We develop a capacity theory based on these spaces. We study basic properties of capacity and several convergence results. As an application, we prove that each weighted variable exponent-Sobolev function has a quasi-continuous representative, we study different definitions of the first order weighted variable exponent-Sobolev spaces with zero boundary values, we define the Dirichlet energy and we prove that it has a minimizer in the weighted variable exponent -Sobolev spaces case.