Related papers: Lebesgue decomposition via Riesz orthogonal decomp…

The aim of this short note is to present an elementary, self-contained, and direct proof for the classical Lebesgue decomposition theorem.

We consider a little-known abstract decomposition result for positive measures due to Dellacherie, and show that it yields many decompositions of measures, several of which are new. We then extend Dellacherie's result to (controlled) vector…

In this paper a new general approach is developed to construct and study Lebesgue type decompositions of linear operators $T$ in the Hilbert space setting. The new approach allows to introduce an essentially wider class of Lebesgue type…

We introduce vector space norms associated to the Mahler measure by using the L^p norm versions of the Weil height recently introduced by Allcock and Vaaler. In order to do this, we determine orthogonal decompositions of the space of…

We give a sketch of proof that any two (Lebesgue) measurable subsets of the unit sphere in $R^n$, for $n\ge 3$, with non-empty interiors and of the same measure are equidecomposable using pieces that are measurable.

The Riesz-Markov theorem identifies any positive, finite, and regular Borel measure on the complex unit circle with a positive linear functional on the continuous functions. By the Weierstrass approximation theorem, the continuous functions…

A linear relation, i.e., a multivalued operator $T$ from a Hilbert space ${\mathfrak H}$ to a Hilbert space ${\mathfrak K}$ has Lebesgue type decompositions $T=T_{1}+T_{2}$, where $T_{1}$ is a closable operator and $T_{2}$ is an operator or…

We establish new and different kinds of proofs of properties that arise due to the orthogonal decomposition of the Hilbert space, including projections, over the unit interval of one dimension. We also see angles between functions,…

The aim of this paper is to prove a general Lebesgue decomposition theorem for positive operators on so-called anti-dual pairs, following the iterative approach introduced by Arlinskii. This procedure and the resulting theorem encompass…

Based on a generalization of Lebesgue decomposition we obtain a characterization of weak compactness in the space $ba$, a representation of its dual space and some results on the structure of finitely additive measures.

The canonical decomposition of a Lorentz algebra element into a sum of orthogonal simple (decomposable) Lorentz bivectors is discussed, as well as the decomposition of a proper orthochronous Lorentz transformation into a product of…

We prove a version of the Lebesgue Differentiation Theorem for mappings that are defined on a measure space and take values into a metric space, with respect to the differentiation basis induced by a von Neumann lifting. As a consequence,…

We generalize the classical de Rham decomposition theorem for Riemannian manifolds to the setting of geodesic metric spaces of finite dimension.

Given all (finite) moments of two measures $\mu$ and $\lambda$ on $\R^n$, we provide a numerical scheme to obtain the Lebesgue decomposition $\mu=\nu+\psi$ with $\nu\ll\lambda$ and $\psi\perp\lambda$. When$\nu$ has a density in…

In this paper, we present a general principle for the Lebesgue measure theory of limsup sets defined by rectangles under the hypothesis of ubiquity for rectangles.

We study the classical problem of identifying the structure of $P^2(\mu)$, the closure of analytic polynomials in the Lebesgue space $L^2(\mu)$ of a compactly supported Borel measure $\mu$ living in the complex plane. In his influential…

This text grew out of notes I have used in teaching a one quarter course on integration at the advanced undergraduate level. My intent is to introduce the Lebesgue integral in a quick, and hopefully painless, way and then go on to…

We present a modification of Riesz's construction of the Lebesgue integral, leading directly to finite or infinite integrals, at the same time simplifying the proofs.

Rademacher theorem states that every Lipschitz function on the Euclidean space is differentiable almost everywhere, where "almost everywhere" refers to the Lebesgue measure. In this paper we prove a differentiability result of similar type,…

We give a new proof the arithmetic Hilbert-Samuel theorem by using classical reductions in the theory of coherent sheaves, a direct proof in the case of the projective space and the conservation of some numerical invariants, called…