Distortion of fuzzy measures is discussed. A special attention is paid to the preservation of submodularity and supermodularity, belief and plausibility. Full characterization of distortion functions preserving the mentioned properties of fuzzy measures is given.
Let be a locally compact group. Let be the left translation in , given by . We characterize (undre a mild set-theoretical hypothesis) the functions such that the map from into is scalarly measurable (i.e. for , is measurable). We show that it is the case when is measurable for each character , and if is compact, if and only if is Riemann-measurable. We show that is Borel measurable if and only if is left uniformly continuous.Some of the measure-theoretic tools used there...
Let be a metric space with a doubling measure, be a boundedly compact metric space and be a Lebesgue precise mapping whose upper gradient belongs to the Lorentz space , . Let be a set of measure zero. Then for -a.e. , where is the -dimensional Hausdorff measure and is the -codimensional Hausdorff measure. This property is closely related to the coarea formula and implies a version of the Eilenberg inequality. The result relies on estimates of Hausdorff content of level sets...
The problem of compact factors in ergodic theory and its relationship with the problem of extending a cocycle to a cocycle of a larger action are studied.
Ces notes ont pour but de rassembler les différents résultats de combinatoire des mots relatifs au billard polygonal et polyédral. On commence par rappeler quelques notions de combinatoire, puis on définit le billard, les notions utiles en dynamique et le codage de l’application. On énonce alors les résultats connus en dimension deux puis trois.
Those who follow Harold Jeffreys in using improper priors together with likelihoods to determine posteriors have thought of the improper measures as probability measures of a deviant sort. This is a mistake. Probability measures are finite measures. Improper distributions generate σ-finite measures. (...)