A note on Riemann integrability

In this note we define Riemann integrabillty for real valued functions defined on a compact metric space accompanied by a finite Borel measure. If the measure of each open ball equals the measure of its corresponding closed ball, then a bounded function is Riemann integrable if and only if its set of points of discontinuity has measure zero.

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On Unique and Nonunique Fixed Points and Fixed Circles in M v b -Metric Space and Application to Cantilever Beam Problem

Journal of Function Spaces ◽

10.1155/2021/6681044 ◽

2021 ◽

Vol 2021 ◽

pp. 1-15

Author(s):

Meena Joshi ◽

Anita Tomar ◽

Hossam A. Nabwey ◽

Reny George

Keyword(s):

Differential Equation ◽

Fixed Points ◽

Metric Space ◽

Order Differential Equation ◽

Unique Fixed Point ◽

Elastic Beam ◽

Closed Ball ◽

Open Ball ◽

Ball Convergence ◽

Fourth Order Differential Equation

We introduce M v b -metric to generalize and improve M v -metric and unify numerous existing distance notions. Further, we define topological notions like open ball, closed ball, convergence of a sequence, Cauchy sequence, and completeness of the space to discuss topology on M v b -metric space and to create an environment for the survival of a unique fixed point. Also, we introduce a notion of a fixed circle and a fixed disc to study the geometry of the set of nonunique fixed points of a discontinuous self-map and establish fixed circle and fixed disc theorems. Further, we verify all these results by illustrative examples to demonstrate the authenticity of the postulates. Towards the end, we solve a fourth order differential equation arising in the bending of an elastic beam.

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Finite powers of strong measure zero sets

Journal of Symbolic Logic ◽

10.2307/2586631 ◽

1999 ◽

Vol 64 (3) ◽

pp. 1295-1306 ◽

Cited By ~ 8

Author(s):

Marion Scheepers

Keyword(s):

Metric Space ◽

Metric Spaces ◽

Measure Zero ◽

General Context ◽

Partition Relation ◽

Compact Metric Space ◽

Totally Bounded ◽

Zero Sets ◽

Strong Measure ◽

Strong Measure Zero

AbstractIn a previous paper—[17]—we characterized strong measure zero sets of reals in terms of a Ramseyan partition relation on certain subspaces of the Alexandroff duplicate of the unit interval. This framework gave only indirect access to the relevant sets of real numbers. We now work more directly with the sets in question, and since it costs little in additional technicalities, we consider the more general context of metric spaces and prove:1. If a metric space has a covering property of Hurewicz and has strong measure zero, then its product with any strong measure zero metric space is a strong measure zero metric space (Theorem 1 and Lemma 3).2. A subspace X of a σ-compact metric space Y has strong measure zero if, and only if, a certain Ramseyan partition relation holds for Y (Theorem 9).3. A subspace X of a σ-compact metric space Y has strong measure zero in all finite powers if, and only if, a certain Ramseyan partition relation holds for Y (Theorem 12).Then 2 and 3 yield characterizations of strong measure zeroness for σ-totally bounded metric spaces in terms of Ramseyan theorems.

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Measurable distal and topological distal systems

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385799133911 ◽

1999 ◽

Vol 19 (4) ◽

pp. 1063-1076 ◽

Cited By ~ 7

Author(s):

ELON LINDENSTRAUSS

Keyword(s):

Continuous Function ◽

Compact Group ◽

Metric Space ◽

Measurable Function ◽

Borel Measure ◽

Base Space ◽

Compact Metric Space ◽

Full Support

In this paper we prove that any ergodic measurably distal system can be realized as a minimal topologically distal system with an invariant Borel measure of full support. The proof depends upon a theorem stating that every measurable function from a measurable system with its base space being a compact metric space to a connected compact group is cohomologous to a continuous function.

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Noncompact closed ball in a constructive compact metric space

Journal of Soviet Mathematics ◽

10.1007/bf01084078 ◽

1976 ◽

Vol 6 (4) ◽

pp. 390-394

Author(s):

V. A. Lifshits ◽

V. P. Chernov

Keyword(s):

Metric Space ◽

Closed Ball ◽

Compact Metric Space

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Existence of Continuous Functions That Are One-to-One Almost Everywhere

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-23688 ◽

2016 ◽

Vol 118 (2) ◽

pp. 269 ◽

Cited By ~ 1

Author(s):

Alexander J. Izzo

Keyword(s):

Metric Space ◽

Borel Measure ◽

Measure Zero ◽

Continuous Functions ◽

Almost Everywhere ◽

Regular Borel Measure ◽

One To One

It is shown that given a metric space $X$ and a $\sigma$-finite positive regular Borel measure $\mu$ on $X$, there exists a bounded continuous real-valued function on $X$ that is one-to-one on the complement of a set of $\mu$ measure zero.

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Strong submeasures and applications to non-compact dynamical systems

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.132 ◽

2020 ◽

pp. 1-23

Author(s):

TUYEN TRUNG TRUONG

Keyword(s):

Metric Space ◽

Bounded Operator ◽

Continuous Functions ◽

Open Dense Subset ◽

Compact Metric Space ◽

Basic Properties ◽

Banach Theorem ◽

Complex Varieties ◽

Definition Of ◽

Meromorphic Maps

Abstract A strong submeasure on a compact metric space X is a sub-linear and bounded operator on the space of continuous functions on X. A strong submeasure is positive if it is non-decreasing. By the Hahn–Banach theorem, a positive strong submeasure is the supremum of a non-empty collection of measures whose masses are uniformly bounded from above. There are many natural examples of continuous maps of the form $f:U\rightarrow X$ , where X is a compact metric space and $U\subset X$ is an open-dense subset, where f cannot extend to a reasonable function on X. We can mention cases such as transcendental maps of $\mathbb {C}$ , meromorphic maps on compact complex varieties, or continuous self-maps $f:U\rightarrow U$ of a dense open subset $U\subset X$ where X is a compact metric space. For the aforementioned mentioned the use of measures is not sufficient to establish the basic properties of ergodic theory, such as the existence of invariant measures or a reasonable definition of measure-theoretic entropy and topological entropy. In this paper we show that strong submeasures can be used to completely resolve the issue and establish these basic properties. In another paper we apply strong submeasures to the intersection of positive closed $(1,1)$ currents on compact Kähler manifolds.

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On equicontinuous factors of flows on locally path-connected compact spaces

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.104 ◽

2020 ◽

pp. 1-18

Author(s):

NIKOLAI EDEKO

Keyword(s):

Lie Group ◽

Metric Space ◽

Betti Number ◽

Alternative Proof ◽

Simple Consequence ◽

Compact Metric Space ◽

Invariant Functions ◽

Compact Spaces ◽

Path Connected

Abstract We consider a locally path-connected compact metric space K with finite first Betti number $\textrm {b}_1(K)$ and a flow $(K, G)$ on K such that G is abelian and all G-invariant functions $f\,{\in}\, \text{\rm C}(K)$ are constant. We prove that every equicontinuous factor of the flow $(K, G)$ is isomorphic to a flow on a compact abelian Lie group of dimension less than ${\textrm {b}_1(K)}/{\textrm {b}_0(K)}$ . For this purpose, we use and provide a new proof for Theorem 2.12 of Hauser and Jäger [Monotonicity of maximal equicontinuous factors and an application to toral flows. Proc. Amer. Math. Soc.147 (2019), 4539–4554], which states that for a flow on a locally connected compact space the quotient map onto the maximal equicontinuous factor is monotone, i.e., has connected fibers. Our alternative proof is a simple consequence of a new characterization of the monotonicity of a quotient map $p\colon K\to L$ between locally connected compact spaces K and L that we obtain by characterizing the local connectedness of K in terms of the Banach lattice $\textrm {C}(K)$ .

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Topological Completeness of Function Spaces Arising in the Hausdorff Approximation of Functions

Canadian Mathematical Bulletin ◽

10.4153/cmb-1992-058-1 ◽

1992 ◽

Vol 35 (4) ◽

pp. 439-448 ◽

Cited By ~ 3

Author(s):

Gerald Beer

Keyword(s):

Approximation Theory ◽

Metric Space ◽

Polish Space ◽

Complete Metric Space ◽

Approximation Of Functions ◽

Lower Envelopes ◽

Constructive Approximation ◽

Hausdorff Approximation ◽

Set Of Points ◽

Completely Metrizable

AbstractLet X be a complete metric space. Viewing continuous real functions on X as closed subsets of X × R, equipped with Hausdorff distance, we show that C(X, R) is completely metrizable provided X is complete and sigma compact. Following the Bulgarian school of constructive approximation theory, a bounded discontinuous function may be identified with its completed graph, the set of points between the upper and lower envelopes of the function. We show that the space of completed graphs, too, is completely metrizable, provided X is locally connected as well as sigma compact and complete. In the process, when X is a Polish space, we provide a simple answer to the following foundational question: which subsets of X × R arise as completed graphs?

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