### A measure theoretic approach to logical quantification

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We present a new characterization of Lebesgue measurable functions; namely, a function f:[0,1]→ ℝ is measurable if and only if it is first-return recoverable almost everywhere. This result is established by demonstrating a connection between almost everywhere first-return recovery and a first-return process for yielding the integral of a measurable function.

A new criterion of asymptotic periodicity of Markov operators on ${L}^{1}$, established in [3], is extended to the class of Markov operators on signed measures.

Let ${G}_{d}$ denote the isometry group of ${\mathbb{R}}^{d}$. We prove that if G is a paradoxical subgroup of ${G}_{d}$ then there exist G-equidecomposable Jordan domains with piecewise smooth boundaries and having different volumes. On the other hand, we construct a system ${\mathcal{F}}_{d}$ of Jordan domains with differentiable boundaries and of the same volume such that ${\mathcal{F}}_{d}$ has the cardinality of the continuum, and for every amenable subgroup G of ${G}_{d}$, the elements of ${\mathcal{F}}_{d}$ are not G-equidecomposable; moreover, their interiors are not G-equidecomposable...

The monotone expectation is defined as a functional over fuzzy measures on finite sets. The functional is based on Choquet functional over capacities and its more relevant properties are proved, including the generalization of classical mathematical expectation and Dempster's upper and lower expectations of an evidence. In second place, the monotone expectation is used to define measures of fuzzy sets. Such measures are compared with the ones based on Sugeno integral. Finally, we prove a generalization...

In it was shown that a (real) signed measure on a cyclic coarse-grained quantum logic can be extended, as a signed measure, over the entire power algebra. Later () this result was re-proved (and further improved on) and, moreover, the non-negative measures were shown to allow for extensions as non-negative measures. In both cases the proof technique used was the technique of linear algebra. In this paper we further generalize the results cited by extending group-valued measures on cyclic coarse-grained...

There is no constraint on the relation between the Fourier and Hausdorff dimension of a set beyond the condition that the Fourier dimension must not exceed the Hausdorff dimension.