Lebesgue Decomposition Research Articles

Variable-Order Fractional Linear Systems with Distributed Delays—Existence, Uniqueness and Integral Representation of the Solutions

In this work, we study a general class of retarded linear systems with distributed delays and variable-order fractional derivatives of Caputo type. We propose an approach consisting of finding an associated one-parameter family of constant-order fractional systems, which is “almost” equivalent to the considered variable-order system in an appropriate sense. This approach allows us to replace the study of the initial problem (IP) for variable-order fractional systems with the study of an IP for these one-parameter families of constant-order fractional systems. We prove that the initial problem for the variable-order fractional system with a discontinuous initial function possesses a unique continuous solution on the half-axis when the function describing the variable order of differentiation is locally bounded, Lebesgue integrable and has an appropriate decomposition similar to the Lebesgue decomposition of functions with bounded variation. The obtained results lead to the existence and uniqueness of a fundamental matrix for the studied variable-order fractional homogeneous system. As an application of the obtained results, we establish an integral representation of the solutions of the studied IP.

Lebesgue decomposition type theorems for weakly null-additive functions on D-posets

Lebesgue decomposition type theorems for weakly null-additive functions on D-posets

Differentiation of measures on an arbitrary measurable space

Let μ,ν be positive finite measures on an arbitrary measurable space (Ω,F), ν=νa+νs be the Lebesgue decomposition of ν with respect to μ, P be the family of all finite partitions π⊆F of Ω, andfπ(μ):=∑A∈π:μ(A)>01Aν(A)μ(A),π∈P. We recall that (fπ(μ))π∈P→dνadμ in L1(μ), as is (essentially) known. Here we identify(πn)n∈N⊆P such thatfπn(μ)→dνadμμ a.s. as n→∞; in the setting of separable F, a (rather trivial) way to do this was already known. To do all this, we characterise the case of equality in Jensen's conditional inequality (generalising the known case of the standard Jensen inequality), and use this to determine how, given a p-uniformly integrable martingale (fi)i∈I, one can identify a sequence(in)n∈N⊆I such that (fin)n converges in Lp to some f which closes the whole net(fi)i. We also give a new proof of the (already-known) characterisation of p-uniformly integrable martingales, without relying on the martingale a.s. convergence theorem.

Optimal borel measure-valued controls to the viscous Cahn–Hilliard–Oberbeck–Boussinesq phase-field system on two-dimensional bounded domains

We consider an optimal control problem for the two-dimensional viscous Cahn-Hilliard-Ober beck-Boussinesq system with controls that take values in the space of regular Borel measures. The state equation models the interaction between two incompressible non-isothermal viscous fluids. Local distributed controls with constraints are applied in either of the equations governing the dynamics for the concentration, mean velocity, and temperature. Necessary and sufficient conditions characterizing local optimality in terms of the Lagrangian will be demonstrated. These conditions will be obtained through regularity results for the associated adjoint system, a priori estimates for the solutions of the linearized system in a weaker norm compared to that of the state space, and the Lebesgue decomposition of Borel measures.

Słociński–Wold decompositions for row isometries

AbstractSłociński gave sufficient conditions for commuting isometries to have a nice Wold-like decomposition. In this note, we provide analogous results for row isometries satisfying certain commutation relations. Other than known results for doubly commuting row isometries, we provide sufficient conditions for a Wold decomposition based on the Lebesgue decomposition of the row isometries.

Characterization of random variables with stationary digits

AbstractLet $q\ge2$ be an integer, $\{X_n\}_{n\geq 1}$ a stochastic process with state space $\{0,\ldots,q-1\}$ , and F the cumulative distribution function (CDF) of $\sum_{n=1}^\infty X_n q^{-n}$ . We show that stationarity of $\{X_n\}_{n\geq 1}$ is equivalent to a functional equation obeyed by F, and use this to characterize the characteristic function of X and the structure of F in terms of its Lebesgue decomposition. More precisely, while the absolutely continuous component of F can only be the uniform distribution on the unit interval, its discrete component can only be a countable convex combination of certain explicitly computable CDFs for probability distributions with finite support. We also show that $\mathrm{d} F$ is a Rajchman measure if and only if F is the uniform CDF on [0, 1].

Connections of unbounded operators and some related topics: von Neumann algebra case

The Kubo–Ando theory deals with connections for positive bounded operators. On the other hand, in various analysis related to von Neumann algebras, it is impossible to avoid unbounded operators. In this paper, we try to extend a notion of connections to cover various classes of positive unbounded operators (or unbounded objects such as positive forms and weights) appearing naturally in the setting of von Neumann algebras, and we must keep all the expected properties maintained. This generalization is carried out for the following classes: (i) positive [Formula: see text]-measurable operators (affiliated with a semifinite von Neumann algebra equipped with a trace [Formula: see text]), (ii) positive elements in Haagerup’s [Formula: see text]-spaces and (iii) semifinite normal weights on a von Neumann algebra. Investigation on these generalizations requires some analysis (such as certain upper semi-continuity) on decreasing sequences in various classes. Several results in this direction are proved, which may be of independent interest. Ando studied Lebesgue decomposition for positive bounded operators by making use of parallel sums. Here, such decomposition is obtained in the setting of noncommutative (Hilsum) [Formula: see text]-spaces.

On Lebesgue decomposition of $p$-adic distributions

On Lebesgue decomposition of $p$-adic distributions

Lebesgue Decomposition of Non-Commutative Measures

AbstractWe extend the Lebesgue decomposition of positive measures with respect to Lebesgue measure on the complex unit circle to the non-commutative (NC) multi-variable setting of (positive) NC measures. These are positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz $C^{\ast }-$algebra, the $C^{\ast }-$algebra of the left creation operators on the full Fock space. This theory is fundamentally connected to the representation theory of the Cuntz and Cuntz–Toeplitz $C^{\ast }-$algebras; any *−representation of the Cuntz–Toeplitz $C^{\ast }-$algebra is obtained (up to unitary equivalence), by applying a Gelfand–Naimark–Segal construction to a positive NC measure. Our approach combines the theory of Lebesgue decomposition of sesquilinear forms in Hilbert space, Lebesgue decomposition of row isometries, free semigroup algebra theory, NC reproducing kernel Hilbert space theory, and NC Hardy space theory.

Noncommutative Lebesgue decomposition and contiguity with applications in quantum statistics

We herein develop a theory of contiguity in the quantum domain based upon a novel quantum analogue of the Lebesgue decomposition. The theory thus formulated is pertinent to the weak quantum local asymptotic normality introduced in the previous paper [Yamagata, Fujiwara, and Gill, Ann. Statist. 41 (2013) 2197–2217], yielding substantial enlargement of the scope of quantum statistics.

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.

Operators on anti-dual pairs: Lebesgue decomposition of positive operators

In this paper we introduce and study absolute continuity and singularity of positive operators acting on anti-dual pairs. We establish a general theorem that can be considered as a common generalization of various earlier Lebesgue-type decompositions. Different algebraic and topological characterizations of absolute continuity and singularity are supplied and also a complete description of uniqueness of the decomposition is provided. We apply the developed decomposition theory to some concrete objects including operators acting in a rigged Hilbert space, Hermitian forms, representable functionals, and additive set functions.

Budget-constrained optimal insurance with belief heterogeneity

We re-examine the problem of budget-constrained demand for insurance indemnification when the insured and the insurer disagree about the likelihoods associated with the realizations of the insurable loss. For ease of comparison with the classical literature, we adopt the original setting of Arrow (1971), but allow for divergence in beliefs between the insurer and the insured; and in particular for singularity between these beliefs, that is, disagreement about zero-probability events. We do not impose the no sabotage condition on admissible indemnities. Instead, we impose a state-verification cost that the insurer can incur in order to verify the loss severity, which rules out ex post moral hazard issues that could otherwise arise from possible misreporting of the loss by the insured. Under a mild consistency requirement between these beliefs that is weaker than the Monotone Likelihood Ratio (MLR) condition, we characterize the optimal indemnity and show that it has a simple two-part structure: full insurance on an event to which the insurer assigns zero probability, and a variable deductible on the complement of this event, which depends on the state of the world through a likelihood ratio. The latter is obtained from a Lebesgue decomposition of the insured’s belief with respect to the insurer’s belief.

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.

Absolute continuity and α-numbers on the real line

Let $\mu,\nu$ be Radon measures on $\mathbb{R}$, with $\mu$ non-atomic and $\nu$ doubling, and write $\mu = \mu_{a} + \mu_{s}$ for the Lebesgue decomposition of $\mu$ relative to $\nu$. For an interval $I \subset \mathbb{R}$, define $\alpha_{\mu,\nu}(I) := \mathbb{W}_{1}(\mu_{I},\nu_{I})$, the Wasserstein distance of normalised blow-ups of $\mu$ and $\nu$ restricted to $I$. Let $\mathcal{S}_{\nu}$ be the square function $$\mathcal{S}^{2}_{\nu}(\mu) = \sum_{I \in \mathcal{D}} \alpha_{\mu,\nu}^{2}(I)\chi_{I},$$ where $\mathcal{D}$ is the family of dyadic intervals of side-length at most one. I prove that $\mathcal{S}_{\nu}(\mu)$ is finite $\mu_{a}$ almost everywhere, and infinite $\mu_{s}$ almost everywhere. I also prove a version of the result for a non-dyadic variant of the square function $\mathcal{S}_{\nu}(\mu)$. The results answer the simplest "$n = d = 1"$ case of a problem of J. Azzam, G. David and T. Toro.

Lebesgue type decompositions for linear relations and Ando’s uniqueness criterion

A linear relation, i.e., a multivalued operator T from a Hilbert space h to a Hilbert space k has Lebesgue type decompositions T = T-1 + T-2, where T i is a closable operator and T-2 is an operator or relation which is singular. There is one canonical decomposition, called the Lebesgue decomposition of T, whose closable part is characterized by its maximality among all closable parts in the sense of domination. All Lebesgue type decompositions are parametrized, which also leads to necessary and sufficient conditions for the uniqueness of such decompositions. Similar results are given for weak Lebesgue type decompositions, where T-1 is just an operator without being necessarily closable. Moreover, closability is characterized in different useful ways. In the special case of range space relations the above decompositions may be applied when dealing with pairs of (nonnegative) bounded operators and nonnegative forms as well as in the classical framework of positive measures.

A Lebesgue-type decomposition for non-positive sesquilinear forms

A Lebesgue-type decomposition of a (not necessarily non-negative) sesquilinear form with respect to a non-negative one is studied. This decomposition consists of a sum of three parts: two are dominated by an absolutely continuous form and a singular non-negative one, respectively, and the latter is majorized by the product of an absolutely continuous and a singular non-negative forms. The Lebesgue decomposition of a complex measure is given as application.

Lebesgue decomposition of probability measures:Applications to equivalence, singularity and kriging

We use Lebesgue decomposition of two probability measures on a measurable space to obtain conditions for their equivalence and singularity in terms of the density of the absolutely continuous part of one probability measure with respect to the other. This allows us to obtain simple proofs of Kakutani’s theorem on product measures (Kakutani, 1948) and an extension of the result of Shepp (1966). In addition, using the density form of two finite-dimensional Gaussian measures, we derive analogues of major results on equivalence and singularity (Parzen, 1963; Kallianpur and Oodaira, 1963; Rozanov, 1968) for Gaussian random fields. These can be used to study the interpolation of the spatial data (Stein, 1999).

On pathwise quadratic variation for càdlàg functions

We revisit H. Foellmer's concept of quadratic variation of a cadlag function along a sequence of time partitions and discuss its relation with the Skorokhod topology. We show that in order to obtain a robust notion of pathwise quadratic variation applicable to sample paths of cadlag processes, one must reformulate the definition of pathwise quadratic variation as a limit in Skorokhod topology of discrete approximations along the partition. One then obtains a simpler definition of quadratic variation which implies the Lebesgue decomposition as a result, rather than requiring it as an extra condition.

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.