It was questions about points on the real line that initiated the study of set theory. Points paved the way to point sets and these to ever more abstract sets. And there was more: Reflection on structural properties of point sets not only initiated the study of ordinary sets; it also supplied blueprints for defining extra-ordinary, “large” sets, transcending those provided by standard set theor...

Since the s many authors have studied the dis tribution of the random series Y P n where the signs are are chosen independently with probability and Solomyak proved that for almost every the distribu tion is absolutely continuous with respect to Lebesgue measure In this paper we prove that is even equivalent to Lebesgue measure for almost all Mathematics Subject Classi cation Primary A A A Rese...

To motivate the elaborate machinery of measure theory, it is desirable to have an example of a set which is not measurable in some natural space. The usual example is the Vitali set, obtained by picking one representative from each equivalence class of R induced by the relation x ∼ y iff x − y ∈ Q. The translation-invariance of Lebesgue measure implies that the resulting set is not Lebesgue-mea...

A weakly continuous near-action of a Polish group G on a standard Lebesgue measure space (X,μ) is whirly if for every A ⊆ X of strictly positive measure and every neighbourhood V of identity in G the set V A has full measure. This is a strong version of ergodicity, and locally compact groups never admit whirly actions. On the contrary, every ergodic near-action by a Polish Lévy group in the sen...

The Schmidt-Eckart-Young theorem for matrices states that the optimal rank-r approximation to a matrix is obtained by retaining the first r terms from the singular value decomposition of that matrix. This work considers a generalization of this optimal truncation property to the CANDECOMP/PARAFAC decomposition of tensors and establishes a necessary orthogonality condition. We prove that this co...

Current theories of real numbers, measure, probability, and stochastic processes appear to be selfinconsistent. We exemplify the inconsistencies with Poisson processes, Markov chains, continuous random variables, the Lebesgue measure, and the Cantor-Dedekind theory of real numbers.

We consider the non-parametric estimation of a function that is observed in white noise after convolution with a boxcar, the indicator of an interval (−a, a). In a recent paper Johnstone et al. (2004) have developed a wavelet deconvolution algorithm (called WaveD) that can be used for “certain” boxcar kernels. For example, WaveD can be tuned to achieve near optimal rates over Besov spaces when ...

We prove completeness of the propositional modal logic S4 for the measure algebra based on the Lebesgue-measurable subsets of the unit interval, [0, 1]. In recent talks, Dana Scott introduced a new measure-based semantics for the standard propositional modal language with Boolean connectives and necessity and possibility operators, and ♦. Propositional modal formulae are assigned to Lebesgue-me...