An Elementary Result on Exponential Measure Spaces

1972 ◽  
Vol 15 (2) ◽  
pp. 277-278
Author(s):  
C. Y. Shen
Keyword(s):  
Measure Space ◽  
Measure Zero ◽  
Measure Theory ◽  
Short Note ◽  
Measurable Subset ◽  
Sufficient Condition ◽  
Product Spaces ◽  
Necessary And Sufficient Condition ◽  
Measure Spaces ◽  
Necessary And Sufficient

A simple but useful result in the measure theory for product spaces can be stated as follows:Theorem A. A necessary and sufficient condition that a measurable subset E of X×Y has measure zero is that almost every X-section (or almost every Y-section) has measure zero (see [1, §36]).We will show, in this short note, that a similar result also holds for the exponential of measure spaces. Before proceeding any further, we describe briefly here the exponential construction of a measure space.

EGOROFF'S THEOREM ON MONOTONE NON-ADDITIVE MEASURE SPACES

2004 ◽  
Vol 12 (01) ◽  
pp. 61-68 ◽  
Author(s):  
JUN LI ◽  
MASAMI YASUDA
Keyword(s):  
Measure Space ◽  
Measure Theory ◽  
Sufficient Condition ◽  
Additive Measure ◽  
Necessary And Sufficient Condition ◽  
Converse Problem ◽  
Almost Everywhere ◽  
Measure Spaces ◽  
Necessary And Sufficient ◽  
Function Sequence

In this paper, the well-known Egoroff's theorem in classical measure theory is established on monotone non-additive measure spaces. Taylor's theorem, which concerns almost everywhere convergence of measurable function sequence in classical measure theory, is also generalized. The converse problem of the theorems are discussed, and a necessary and sufficient condition for the Egoroff's theorem is obtained on semicontinuous fuzzy measure space with S-compactness.

scholarly journals Bakry-Émery Conditions on Almost Smooth Metric Measure Spaces

2018 ◽  
Vol 6 (1) ◽  
pp. 129-145 ◽  
Author(s):  
Shouhei Honda
Keyword(s):  
Riemannian Manifolds ◽  
Short Note ◽  
Sufficient Condition ◽  
Metric Measure Spaces ◽  
Necessary And Sufficient Condition ◽  
Local Dimension ◽  
Base Points ◽  
Smooth Metric Measure Spaces ◽  
Measure Spaces ◽  
Necessary And Sufficient

Abstract In this short note, we give a sufficient condition for almost smooth compact metric measure spaces to satisfy the Bakry-Émery condition BE(K, N). The sufficient condition is satisfied for the glued space of any two (not necessary same dimensional) closed pointed Riemannian manifolds at their base points. This tells us that the BE condition is strictly weaker than the RCD condition even in this setting, and that the local dimension is not constant even if the space satisfies the BE condition with the coincidence between the induced distance by the Cheeger energy and the original distance. In particular, the glued space gives a first example with a Ricci bound from below in the Bakry-Émery sense, whose local dimension is not constant. We also give a necessary and sufficient condition for such spaces to be RCD(K, N) spaces.

A SUPER RADON-NIKODYM DERIVATIVE FOR ALMOST SUBADDITIVE SET FUNCTIONS

2013 ◽  
Vol 21 (03) ◽  
pp. 347-365 ◽  
Author(s):  
YANN RÉBILLÉ
Keyword(s):  
Density Function ◽  
Measure Theory ◽  
Type Theorem ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Set Functions ◽  
Necessary And Sufficient ◽  
Exact Factorization ◽  
Nikodym Derivative ◽  
Nikodym Theorem

In classical measure theory, the Radon-Nikodym theorem states in a concise condition, namely domination, how a measure can be factorized by another (bounded) measure through a density function. Several approaches have been undertaken to see under which conditions an exact factorization can be obtained with set functions that are not σ-additive (for instance finitely additive set functions or submeasures). We provide a Radon-Nikodym type theorem with respect to a measure for almost subadditive set functions with bounded disjoint variation. The necessary and sufficient condition to guarantee a superior Radon-Nikodym derivative remains the standard domination condition for measures. We show how these set functions admit an equivalent factorization under the standard domination condition for set functions.

Strongly equivalent invariant measures

1980 ◽  
Vol 88 (1) ◽  
pp. 33-43 ◽  
Author(s):  
J. Rosenblatt
Keyword(s):  
Measure Space ◽  
Positive Measure ◽  
Invariant Measures ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Zero Measure ◽  
Infinite Measure ◽  
Two Measures ◽  
Necessary And Sufficient

AbstractTwo measures are strongly equivalent if they have the same sets of zero measure and the same sets of infinite measure. Given a group G of strongly non-singular measurable transformations of a non-atomic positive measure space (X, β, p), if G is amenable, then a necessary and sufficient condition for there to be a G-invariant positive measure on (X, β) which is strongly equivalent to p is that p(E) > 0 implies inf p(gE) > 0 and also p(E) < ∞ implies

A continuous generalization of the transversal property

1984 ◽  
Vol 95 (1) ◽  
pp. 21-23
Author(s):  
P. Komjáth
Keyword(s):  
Finite Subset ◽  
Measure Space ◽  
Choice Function ◽  
Finite Measure ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finite Sets ◽  
Real Numbers ◽  
One To One ◽  
Necessary And Sufficient

A transversal for a set-system is a one-to-one choice function. A necessary and sufficient condition for the existence of a transversal in the case of finite sets was given by P. Hall (see [4, 3]). The corresponding condition for the case when countably many countable sets are given was conjectured by Nash-Williams and later proved by Damerell and Milner [2]. B. Bollobás and N. Varopoulos stated and proved the following measure theoretic counterpart of Hall's theorem: if (X, μ) is an atomless measure space, ℋ = {Hi: i∈I} is a family of measurable sets with finite measure, λi (i∈I) are non-negative real numbers, then we can choose a subset Ti ⊆ Hi with μ(Ti) = λi and μ(Ti ∩ Ti′) = 0 (i ≠ i′) if and only if μ({U Hi: iεJ}) ≥ Σ{λi: iεJ}: for every finite subset J of I. In this note we generalize this result giving a necessary and sufficient condition for the case when I is countable and X is the union of countably many sets of finite measure.

scholarly journals Acceptable Morita Contexts for Semigroups

ISRN Algebra ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-5 ◽  
Author(s):  
Valdis Laan
Keyword(s):  
Short Note ◽  
Morita Equivalence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Morita Context ◽  
Necessary And Sufficient

This short note deals with Morita equivalence of (arbitrary) semigroups. We give a necessary and sufficient condition for a Morita context containing two semigroups S and T to induce an equivalence between the category of closed right S-acts and the category of closed right T-acts.

scholarly journals Local monotonicity coefficients in Orlicz sequence spaces equipped with the p-Amemiya norm

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xin He ◽  
Yunan Cui ◽  
Henryk Hudzik
Keyword(s):  
Measure Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Orlicz Sequence Space ◽  
Atomic Measure ◽  
Uniform Monotonicity ◽  
Orlicz Sequence Spaces ◽  
Necessary And Sufficient ◽  
Local Monotonicity ◽  
Amemiya Norm

Abstract In this paper, the monotonicity is investigated with respect to Orlicz sequence space $l_{\varPhi , p}$ l Φ , p equipped with the p-Amemiya norm, and the necessary and sufficient condition is obtained to guarantee the uniform monotonicity, locally uniform monotonicity, and strict monotonicity for $l_{\varPhi , p}$ l Φ , p . This completes the results of the paper (Cui et al. in J. Math. Anal. Appl. 432:1095–1105, 2015) which were obtained for the non-atomic measure space. Local upper and lower coefficients of monotonicity at any point of the unit sphere are calculated, $l_{\varPhi , p}$ l Φ , p is calculated.

scholarly journals A Characterization of the Intuitionistic Propositional Logic

1970 ◽  
Vol 37 ◽  
pp. 131-136
Author(s):  
Nobol Muti
Keyword(s):  
Propositional Logic ◽  
Short Note ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Intuitionistic Propositional Logic ◽  
Necessary And Sufficient

In this short note, is shown a necessary and sufficient condition for a logic to be an intermediate propositional logic in Umezawa’s sense (see the reference), under such an assumption that any logic in consideration (as a subclass of LK-provable propositions) contains at least the axioms of the positive propositional logic LPS (Curry’s LA) as its axioms and is closed with respect to the rules of detachment and substitution.

scholarly journals C*-Algebras of Irreversible Dynamical Systems

2006 ◽  
Vol 58 (1) ◽  
pp. 39-63 ◽  
Author(s):  
R. Exel ◽  
A. Vershik
Keyword(s):  
Measure Space ◽  
General Theorem ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Circle Actions ◽  
Generators And Relations ◽  
C Algebras ◽  
Special Cases ◽  
Measure Preserving Transformation ◽  
Necessary And Sufficient

AbstractWe show that certain C*-algebras which have been studied by, among others, Arzumanian, Vershik, Deaconu, and Renault, in connection with a measure-preserving transformation of a measure space or a covering map of a compact space, are special cases of the endomorphism crossed product construction recently introduced by the first named author. As a consequence these algebras are given presentations in terms of generators and relations. These results come as a consequence of a general theorem on faithfulness of representations which are covariant with respect to certain circle actions. For the case of topologically free covering maps we prove a stronger result on faithfulness of representations which needs no covariance. We also give a necessary and sufficient condition for simplicity.

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