Lebesgue Decomposition Theorem Research Articles

Bilateral Risk Sharing with Heterogeneous Beliefs and Exposure Constraints

This paper studies bilateral risk sharing under no aggregate uncertainty, where one agent has Expected-Utility preferences and the other agent has Rank-Dependent Utility preferences with a general probability distortion function. We impose exogenous constraints on the risk exposure for both agents, and we allow for any type or level of belief heterogeneity. We show that Pareto-optimal risk-sharing contracts can be obtained via a constrained utility maximization under a participation constraint of the other agent. This allows us to give an explicit characterization of optimal risk-sharing contracts. In particular, we show that an optimal risk-sharing contract contains allocations that are monotone functions of the likelihood ratio, where the latter is obtained from Lebesgue's Decomposition Theorem.

Hahn’s Proof of the Hahn Decomposition Theorem, and Related Matters

The easiest proof of the Hahn decomposition theorem for signed measure spaces is Hahn’s own proof, yet it has been forgotten. This note aims to redress that and also aims to draw attention to some facts about the Radon–Nikodym theorem and the Lebesgue decomposition theorem that should be better known than they are.

The Singular Part as Fixed Point

The aim of this note is to investigate the Lebesgue decomposition theorem of nonnegative finite measures from a new point of view. Using a standard iteration scheme, we identify the singular part as a fixed point of a nonnegative finite measure-valued map.

Existence Theorems for a Multidimensional Crystal Surface Model

In this paper we study the existence assertion of the initial boundary value problem for the equation $\frac{\partial u}{\partial t} = \Delta e^{-\Delta u}$. This problem arises in the mathematical description of the evolution of crystal surfaces. Our investigations reveal that the exponent in the equation can have a singular part in the sense of the Lebesgue decomposition theorem, and the exponential nonlinearity somehow “cancels” it out. The net result is that we obtain a solution $u$ that satisfies the equation and the initial boundary conditions in the almost everywhere (a.e.) sense.

A Simple Proof of the Lebesgue Decomposition Theorem

Motivated by the notion of operator, a short and simple proof of Lebesgue's decomposition theorem is presented in this note.

LEBESGUE DECOMPOSITION FOR REPRESENTABLE FUNCTIONALS ON *-ALGEBRAS

AbstractWe offer a Lebesgue-type decomposition of a representable functional on a *-algebra into absolutely continuous and singular parts with respect to another. Such a result was proved by Zs. Szűcs due to a general Lebesgue decomposition theorem of S. Hassi, H.S.V. de Snoo, and Z. Sebestyén concerning non-negative Hermitian forms. In this paper, we provide a self-contained proof of Szűcs' result and in addition we prove that the corresponding absolutely continuous parts are absolutely continuous with respect to each other.

Absolute continuity of positive linear functionals

The goal of this paper is to present characterizations for absolute continuity of representable positive functionals on general $^*$-algebras. From the results we give a new and very different proof to our recently published Lebesgue decomposition theorem for representable positive functionals. On unital $C^*$-algebras and measure algebras of compact groups further characterizations are included in the paper. As an application of our results, we answer Gudder's problem on the uniqueness of the Lebesgue decomposition in the case of commutative $^*$-algebras and measure algebras of compact groups. Another application to faithful positive functionals defined on the latter $^*$-algebras is also included.

On the Lebesgue decomposition for non-additive functions

A Lebesgue decomposition theorem for non-additive functions, acting on a $$\sigma $$ -complete orthomodular lattice and taking values in Hausdorff uniform spaces, is established. No algebraic structure is required on target spaces. The Boolean case is also investigated.

Lebesgue decomposition theorems

In the present paper we offer an operator theoretic proof of the Lebesgue decomposition theorem among nonnegative forms on (real or complex vector spaces. The basic tool in our treatment is the embedding operator between two auxiliary Hilbert spaces associated to the forms in question. As an application of our approach, we also provide the Lebesgue decomposition theorem among finitely additive bounded set functions on rings of sets.

Decomposition of ℓ-group-valued measures

We deal with decomposition theorems for modular measures µ: L → G defined on a D-lattice with values in a Dedekind complete ℓ-group. Using the celebrated band decomposition theorem of Riesz in Dedekind complete ℓ-groups, several decomposition theorems including the Lebesgue decomposition theorem, the Hewitt-Yosida decomposition theorem and the Alexandroff decomposition theorem are derived. Our main result—also based on the band decomposition theorem of Riesz—is the Hammer-Sobczyk decomposition for ℓ-group-valued modular measures on D-lattices. Recall that D-lattices (or equivalently lattice ordered effect algebras) are a common generalization of orthomodular lattices and of MV-algebras, and therefore of Boolean algebras. If L is an MV-algebra, in particular if L is a Boolean algebra, then the modular measures on L are exactly the finitely additive measures in the usual sense, and thus our results contain results for finitely additive G-valued measures defined on Boolean algebras.

A Radon–Nikodym type theorem for forms

The Lebesgue decomposition theorem and the Radon–Nikodym theorem are the cornerstones of the classical measure theory. These theorems were generalized in several settings and several ways. Hassi, Sebestyen, and de Snoo recently proved a Lebesgue type decomposition theorem for nonnegative sesquilinear forms defined on complex linear spaces. The main purpose of this paper is to formulate and prove also a Radon–Nikodym type result in this setting. As an application, we present a Lebesgue type decomposition theorem and solve a special case of the infimum problem for densely defined (not necessarily bounded) positive operators.

First order tameness of measures

We develop a general framework for measure theory and integration theory that is compatible with o-minimality. Therefore the following natural definitions are introduced. Given are an o-minimal structure M and a Borel measure μ on some Rn. We say that μ is M-compatible if there is an o-minimal expansion of M such that for every parameterized family of subsets of Rn that is definable in M the corresponding family of μ-measures is definable in this o-minimal expansion. We say that μ is M-tame if there is an o-minimal expansion of M such that for every parameterized family of functions on Rn that is definable in M the corresponding family of integrals with respect to μ is definable in this o-minimal expansion. We substantiate these definitions with existing and many new examples. We investigate the Lebesgue measure in their light. We prove definable versions of fundamental results such as the theorem of Radon–Nikodym, the Lebesgue decomposition theorem and the Riesz representation theorem. They allow us to describe explicitly compatible and tame measures and to classify them in dimension one.

The Lebesgue Decomposition Theorem for Arbitrary Contents

The decomposition theorem named after Lebesgue asserts that certain set functions have canonical representations as sums of particular set functions called the absolutely continuous and the singular ones with respect to some fixed set function. The traditional versions are for the bounded measures with respect to some fixed measure on a σ algebra, in final form due to Hahn 1921, and for the bounded contents with respect to some fixed content on an algebra, due to Bochner-Phillips 1941 and Darst 1962. Then came the version for arbitrary measures, due to R.A.Johnson 1967 and N.Y.Luther 1968. The unpleasant fact with these versions is that each one requires its particular notions of absolutely continuous and singular constituents. It seems mysterious how a common roof for all of them could look, and therefore how a universal version for arbitrary contents could be achieved - and all that while several abstract extensions of particular versions appeared in the subsequent decades, for example due to de Lucia-Morales 2003. After these decades now the present article claims to arrive at the final aim in the original context of arbitrary contents. The article will be based on the author’s new difference formation for arbitrary contents 1999. This difference formation even furnishes simple explicit formulas for the two constituents.

Lebesgue decomposition theorem for σ-finite signed fuzzy measures

Further research on signed fuzzy measures is made. Lebesgue decomposition theorems for σ-finite signed fuzzy measures and fuzzy measures are proved under the null-null additive condition. Thus, the relative result in classical measure theory is generalized.

Lebesgue decomposition by σ-null-additive set functions using ideals

Using V. Ficker’s and P. Capek’s algebraic approach by ideals a Lebesgue decomposition theorem for a wide class of non-additive set functions called null-additive set functions is obtained. In special cases decompositions theorems for ⊕-decomposable measures andk-triangular set functions are obtained.

The Lebesgue decomposition theorem for fuzzy quantum spaces

Abstract The present paper is devoted to signed measures of fuzzy quantum spaces. The Lebesque decomposition theorem is proved for the fuzzy case.

Decompositions of supermodular functions and □-decomposable measures

Abstract We consider supermodular functions and □-decomposable measures defined on σ-complete lattice and with values in σ-complete lattice ordered semigroup. For such supermodular functions and □-decomposable measures on lattice with relative complement Hewitt-Yosida decomposition type theorem and Lebesgue decomposition theorem are proved.

Regularity properties of non-additive set functions

Abstract We investigate some regularity properties of non-additive set functions; a Lebesgue decomposition theorem is proved for measures that are decomposable with regard to a triangular conorm.

A wide Perron integral

In terms of an arbitrary limit process T, defined abstractly for real functions, we define in a novel way a T-continuous integral of Perron type, admitting mean value theorems, integration by parts and the analogue of the Marcinkiewicz theorem for the ordinary Perron integral. The integral is shown to include, as particular cases, the various known continuous, approximately continuous, cesàro-continuous, mean-continuous and proximally Cesàro-continuous integrals of Perron and Denjoy types. An interesting generalization of the classical Lebesgue decomposition theorem is also obtained.

Decompositions of Submeasures

In [4] we showed that one can tell whether a submeasure on a Boolean algebra has a control measure or is pathological by comparing the Fréchet-Nikodym topology it generates to the universal measure topology of Graves. We then wondered if a submeasure could be decomposed into a part with a control measure and a part which is pathological or zero. This led to the problem of finding a Lebesgue decomposition for a submeasure on an algebra of sets with respect to a Fréchet-Nikodym topology.In [6] Drewnowski proved a Lebesgue decomposition theorem for exhaustive submeasures with respect to “additivities” and a similar theorem for exhaustive Fréchet-Nikodym topologies. He asked if an exhaustive Fréchet-Nikodym topology could be decomposed with respect to another Fréchet-Nikodym topology. In [12] Traynor showed that the answer is “yes”.