The Caratheodory Extension Theorem for Vector Valued Measures

1978 ◽  
Vol 72 (1) ◽  
pp. 57 ◽  
Author(s):  
Joseph Kupka
Keyword(s):  
Extension Theorem ◽  
Carathéodory Extension Theorem ◽  
Vector Valued

scholarly journals Approximation and extension of continuous functions

1994 ◽  
Vol 57 (2) ◽  
pp. 149-157 ◽  
Author(s):  
Salvador Hernández-Muñoz
Keyword(s):  
Topological Space ◽  
Short Proof ◽  
Extension Theorem ◽  
Continuous Functions ◽  
Vector Valued

AbstractIn this paper we study the approximation of vector valued continuous functions defined on a topological space and we apply this study to different problems. Thus we give a new proof of Machado's Theorem. Also we get a short proof of a Theorem of Katětov and we prove a generalization of Tietze's Extension Theorem for vector-valued continuous functions, thereby solving a question left open by Blair.

scholarly journals Mean value theorems and a Taylor theorem for vector valued functions

1981 ◽  
Vol 24 (1) ◽  
pp. 69-77
Author(s):  
Rudolf Výborný
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Extension Theorem ◽  
Mean Value ◽  
Mean Value Theorems ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Locally Convex

Two mean value theorems and a Taylor theorem for functions with values in a locally convex topological vector space are proved without the use of the Hahn-Banach extension theorem.

scholarly journals Generalized Caratheodory Extension Theorem on Fuzzy Measure Space

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Mehmet Şahin ◽  
Necati Olgun ◽  
F. Talay Akyıldız ◽  
Ali Karakuş
Keyword(s):  
Measure Space ◽  
Extension Theorem ◽  
Fuzzy Measure ◽  
Outer Measure ◽  
Fuzzy Measures ◽  
Set Functions ◽  
Bottom Element ◽  
Entire Universe ◽  
Carathéodory Extension Theorem

Lattice-valued fuzzy measures are lattice-valued set functions which assign the bottom element of the lattice to the empty set and the top element of the lattice to the entire universe, satisfying the additive properties and the property of monotonicity. In this paper, we use the lattice-valued fuzzy measures and outer measure definitions and generalize the Caratheodory extension theorem for lattice-valued fuzzy measures.

scholarly journals Quasiconformal Jordan Domains

2021 ◽  
Vol 9 (1) ◽  
pp. 167-185
Author(s):  
Toni Ikonen
Keyword(s):  
Metric Space ◽  
Jordan Domain ◽  
Extension Theorem ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Quasiconformal Homeomorphism ◽  
Carathéodory Extension Theorem ◽  
Necessary And Sufficient

Abstract We extend the classical Carathéodory extension theorem to quasiconformal Jordan domains (Y, dY ). We say that a metric space (Y, dY ) is a quasiconformal Jordan domain if the completion ̄Y of (Y, dY ) has finite Hausdorff 2-measure, the boundary ∂Y = ̄Y \ Y is homeomorphic to 𝕊1, and there exists a homeomorphism ϕ: 𝔻 →(Y, dY ) that is quasiconformal in the geometric sense. We show that ϕ has a continuous, monotone, and surjective extension Φ: 𝔻 ̄ → Y ̄. This result is best possible in this generality. In addition, we find a necessary and sufficient condition for Φ to be a quasiconformal homeomorphism. We provide sufficient conditions for the restriction of Φ to 𝕊1 being a quasisymmetry and to ∂Y being bi-Lipschitz equivalent to a quasicircle in the plane.

The Vector-Valued Maximin

1994 ◽  
Keyword(s):  
Vector Valued

Trade-Off and Consensus in Vector-Valued Optimization Theory

2001 ◽  
Vol 33 (9) ◽  
pp. 10
Author(s):  
Albert N. Voronin
Keyword(s):  
Optimization Theory ◽  
Trade Off ◽  
Vector Valued

The range of block Hankel operators

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3237-3243
Author(s):  
In Hwang ◽  
In Kim ◽  
Sumin Kim
Keyword(s):  
Hardy Space ◽  
Invariant Subspace ◽  
Hankel Operators ◽  
Coprime Factorization ◽  
Backward Shift ◽  
Vector Valued

In this note we give a connection between the closure of the range of block Hankel operators acting on the vector-valued Hardy space H2Cn and the left coprime factorization of its symbol. Given a subset F ? H2Cn, we also consider the smallest invariant subspace S*F of the backward shift S* that contains F.

Operators from H1 into a Banach space and vector valued measures

1989 ◽  
pp. 87-93
Author(s):  
Oscar Blasco
Keyword(s):  
Banach Space ◽  
Vector Valued

Generic rotation sets

2020 ◽  
pp. 1-13
Author(s):  
SEBASTIÁN PAVEZ-MOLINA
Keyword(s):  
Dynamical System ◽  
Convex Body ◽  
Probability Measures ◽  
Topological Dynamical System ◽  
Continuous Vector ◽  
Rotation Set ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Uniquely Ergodic ◽  
Dense Set

Abstract Let $(X,T)$ be a topological dynamical system. Given a continuous vector-valued function $F \in C(X, \mathbb {R}^{d})$ called a potential, we define its rotation set $R(F)$ as the set of integrals of F with respect to all T-invariant probability measures, which is a convex body of $\mathbb {R}^{d}$ . In this paper we study the geometry of rotation sets. We prove that if T is a non-uniquely ergodic topological dynamical system with a dense set of periodic measures, then the map $R(\cdot )$ is open with respect to the uniform topologies. As a consequence, we obtain that the rotation set of a generic potential is strictly convex and has $C^{1}$ boundary. Furthermore, we prove that the map $R(\cdot )$ is surjective, extending a result of Kucherenko and Wolf.

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