Related papers: Measures by means, means by measures
The aim of this article is to establish basic results in a conditional measure theory. The results are applied to prove that arbitrary kernels and conditional distributions are represented by measures in a conditional set theory. In…
The operations of linear algebra, calculus, and statistics are routinely applied to measurement scales but certain mathematical conditions must be satisfied in order for these operations to be applicable. We call attention to the conditions…
A classical theorem of Menshov states that every measurable function can redefined on a set of arbitrarily small Lebesgue measure, so that the resulting function has uniformly convergent Fourier series. We prove that the same is true if we…
Given a finite Borel measure $\mu$ on R n and basic semi-algebraic sets $\Omega$\_i $\subset$ R n , i = 1,. .. , p, we provide a systematic numerical scheme to approximate as closely as desired $\mu$(\cup\_i $\Omega$\_i), when all moments…
The Assouad and lower dimensions and dimension spectra quantify the regularity of a measure by considering the relative measure of concentric balls. On the other hand, one can quantify the smoothness of an absolutely continuous measure by…
Based on collection of bijections, variable and function are extended into ``isomorphic variable'' and ``dual-variable-isomorphic function'', then mean values such as arithmetic mean and mean of a function are extended to ``isomorphic…
Skewness measures can be used to measure the level of asymmetry of a distribution. Given the prevalence of statistical methods that assume underlying symmetry, and also the desire for symmetry in order to make meaningful judgements for…
Both classical and respectively quantum observables can be modeled as somewhat similar examples of random variables. In such a model the associated measurements preserve the values spectrum of an observable but change the corresponding…
With a new proof approach we prove in a more general setting the classical convergence theorem that almost everywhere convergence of measurable functions on a finite measure space implies convergence in measure. Specifically, we generalize…
Necessary and sufficient conditions for a measure to be an extreme point of the set of measures (on an abstract measurable space) with prescribed generalized moments are given, as well as an application to extremal problems over such moment…
Let X be a non-empty set and U a ring of subsets of X. The countable additive functions U->{0,1} are called measures. The paper gives some definitions (derivable measures, the Lebesgue-Stieltjes measures) and properties of these functions,…
If $\Lambda $ is a measure space, $u:\Lambda ^{m}\rightarrow \Bbb{R}$ is a given function and $N\geq m,$ the function $U(x_{1},...,x_{N})=\left( \begin{array}{l} N \\ m \end{array} \right) ^{-1}\sum_{1\leq i_{1}<\cdots <i_{m}\leq…
In this paper, we review the basic properties of measures vanishing at infinity and prove a version of the Riemann--Lebesgue lemma for Fourier transformable measures.
We construct a function on the real line supported on a set of finite measure whose spectrum has density zero.
This paper is dedicated to the analysis and detailed study of a procedure to generate both the weighted arithmetic and harmonic means of $n$ positive real numbers. Together with this interpretation, we prove some relevant properties that…
We define a Gaussian invariant measure for the two-dimensional averaged-Euler equation and show the existence of its solution with initial conditions on the support of the measure. An invariant surface measure on the level sets of the…
We study the randomness properties of reals with respect to arbitrary probability measures on Cantor space. We show that every non-computable real is non-trivially random with respect to some measure. The probability measures constructed in…
In this paper we study the problem of extending means to means of higher order. We show how higher order means can be inductively defined and established in general metric spaces, in particular, in convex metric spaces. As a particular…
How should researchers analyze randomized experiments in which the main outcome is latent and measured in multiple ways but each measure contains some degree of error? We first identify a critical study-specific noncomparability problem in…
We consider nonlinear, or "event-dependent", sampling, i.e. such that the sampling instances {tk} depend on the function being sampled. The use of such sampling in the construction of Lebesgue's integral sums is noted and discussed as…