scholarly journals Absolute continuity, singularity and product measures

1982 ◽  
Vol 5 (4) ◽  
pp. 793-807
Author(s):  
Roy A. Johnson
Keyword(s):  
Absolute Continuity ◽  
Decomposition Theorem ◽  
Absolutely Continuous ◽  
Weakly Singular ◽  
Lebesgue Type Decomposition ◽  
Product Measures ◽  
Type Decomposition

Conditions are given under which a product of two semifinite measures is absolutely continuous or weakly singular with respect to another product of two semifinite measures. A Lebesgue type decomposition theorem is proved for certain product measures so that the resulting measures are themselves product measures.

scholarly journals LEBESGUE DECOMPOSITION FOR REPRESENTABLE FUNCTIONALS ON *-ALGEBRAS

2015 ◽  
Vol 58 (2) ◽  
pp. 491-501 ◽  
Author(s):  
ZSIGMOND TARCSAY
Keyword(s):  
Decomposition Theorem ◽  
Lebesgue Decomposition ◽  
Absolutely Continuous ◽  
Hermitian Forms ◽  
Lebesgue Type Decomposition ◽  
Representable Functional ◽  
Type Decomposition

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.

scholarly journals Lebesgue Decomposition Theorem and Weak Radon-Nikodým Theorem for Generalized Fuzzy Number Measures

2015 ◽  
Vol 2015 ◽  
pp. 1-8
Author(s):  
Cai-Li Zhou ◽  
Fu-Gui Shi
Keyword(s):  
Banach Spaces ◽  
Fuzzy Number ◽  
Decomposition Theorem ◽  
Lebesgue Decomposition ◽  
Generalized Fuzzy Number ◽  
Lebesgue Type Decomposition ◽  
Nikodym Theorem ◽  
Type Decomposition

The Lebesgue type decomposition theorem and weak Radon-Nikodým theorem for fuzzy valued measures in separable Banach spaces are established.

A Lebesgue Decomposition Theorem for C* Algebras

1972 ◽  
Vol 15 (1) ◽  
pp. 87-91 ◽  
Author(s):  
Michael Henle
Keyword(s):  
Absolute Continuity ◽  
Linear Functional ◽  
Decomposition Theorem ◽  
Lebesgue Decomposition ◽  
C Algebras ◽  
Positive Linear Functional ◽  
Decomposition Theorems

This paper, by generalizing von Neumann's proof of the Radon-Nikodym and Lebesgue decomposition theorems [3], obtains analogous results for positive linear functional on a C* algebra. The concept of "absolute continuity" used and the Radon-Nikodym portion of the resulting theorem are due to Dye [2].

scholarly journals Center foliation: absolute continuity, disintegration and rigidity

2014 ◽  
Vol 36 (1) ◽  
pp. 256-275 ◽  
Author(s):  
RÉGIS VARÃO
Keyword(s):  
Absolute Continuity ◽  
Geodesic Flows ◽  
Absolutely Continuous ◽  
Rigidity Result ◽  
Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic ◽  
Smooth Conjugacy ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic Diffeomorphism ◽  
Volume Preserving

In this paper we address the issues of absolute continuity for the center foliation, as well as the disintegration on the non-absolute continuous case and rigidity of volume-preserving partially hyperbolic diffeomorphisms isotopic to a linear Anosov automorphism on $\mathbb{T}^{3}$. It is shown that the disintegration of volume on center leaves for these diffeomorphisms may be neither atomic nor Lebesgue, in contrast to the dichotomy (Lebesgue or atomic) obtained by Avila, Viana and Wilkinson [Absolute continuity, Lyapunov exponents and rigidity I: Geodesic flows. Preprint, 2012, arXiv:1110.2365v2] for perturbations of time-one of geodesic flow. In the case of atomic disintegration of volume on the center leaves of an Anosov diffeomorphism on $\mathbb{T}^{3}$, we show that it has to be one atom per leaf. Moreover, we show that not even a $C^{1}$ center foliation implies a rigidity result. However, for a volume-preserving partially hyperbolic diffeomorphism isotopic to a linear Anosov automorphism, assuming the center foliation is $C^{1}$ and transversely absolutely continuous with bounded Jacobians, we obtain smooth conjugacy to its linearization.

Subordinacy Analysis and Absolutely Continuous Spectra for Sturm-Liouville Equations with Two Singular Endpoints

1998 ◽  
Vol 41 (1) ◽  
pp. 23-27
Author(s):  
Dominic P. Clemence
Keyword(s):  
Liouville Equation ◽  
Absolute Continuity ◽  
Absolutely Continuous ◽  
Particular Solution ◽  
Sturm Liouville ◽  
The One

AbstractThe Gilbert-Pearson characterization of the spectrum is established for a generalized Sturm-Liouville equation with two singular endpoints. It is also shown that strong absolute continuity for the one singular endpoint problem guarantees absolute continuity for the two singular endpoint problem. As a consequence, we obtain the result that strong nonsubordinacy, at one singular endpoint, of a particular solution guarantees the nonexistence of subordinate solutions at both singular endpoints.

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