scholarly journals On density with respect to equivalent measures

2010 ◽  
Vol 43 (1) ◽  
Author(s):  
Artur Bartoszewicz ◽  
Małgorzata Filipczak ◽  
Tadeusz Poreda
Keyword(s):  
Metric Space ◽  
Borel Measure ◽  
Borel Subset ◽  
Density Point

AbstractIn the paper there is disscussed a notion of a density point of a Borel subset of a metric space with respect to a Borel measure

scholarly journals Uniform boundedness principles for regular Borel vector measures

1980 ◽  
Vol 29 (2) ◽  
pp. 206-218 ◽  
Author(s):  
Joseph Kupka
Keyword(s):  
Banach Space ◽  
Borel Measure ◽  
Borel Subset ◽  
Uniform Boundedness ◽  
Vector Measures ◽  
Boundedness Theorem ◽  
Regular Borel Measure ◽  
Pointwise Limit ◽  
Regular Open ◽  
Uniformly Bounded

The setting is a compact Hausfroff space ω. The notion of a Walls class of subsets of Ω is defined via strange axioms—axioms whose justification rests with examples such as the collection of regular open sets or the range of a strong lifting. Avarient of Rosenthal' famous lwmma which applies directly to Banach space-valued measures is esablished, and it is used to obtain, in elementary fashion, the following two uniform boundedness principles: (1)The Nikodym Boundedness Theorem. If K is a family of regular Borel vector measures on Ω which is point-wise bounded on every set of a fixed Wells class, then K is uniformly bounded. (2)The Nikodym Covergence Theorem. If {μn} is a sequence of regular Borel vector measures on Ω which is converguent on every set of a fixed Wells class, then the μn are uniformly countably additive, the sequence {μn} is convergent on every Borel subset of Ω and the pointwise limit constitutes a regular Borel measure.

Measurable distal and topological distal systems

1999 ◽  
Vol 19 (4) ◽  
pp. 1063-1076 ◽  
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.

scholarly journals A note on Riemann integrability

1978 ◽  
Vol 1 (1) ◽  
pp. 69-73
Author(s):  
G. A. Beer
Keyword(s):  
Metric Space ◽  
Borel Measure ◽  
Measure Zero ◽  
Bounded Function ◽  
Closed Ball ◽  
Open Ball ◽  
Compact Metric Space ◽  
Finite Borel Measure ◽  
Set Of Points ◽  
Points Of Discontinuity

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.

A Note on Regular Measures

1964 ◽  
Vol 7 (1) ◽  
pp. 41-44 ◽  
Author(s):  
Elias Zakon
Keyword(s):  
Simple Proof ◽  
Metric Space ◽  
Borel Measure ◽  
Metric Spaces ◽  
Finite Measure ◽  
Closed Set ◽  
Countable Intersection

As is well known, every Borel measure in a metric space S is regular, provided that S is the union of a sequence of open sets of finite measure. It seems, however, not yet to have been noticed that this theorem can be easily extended to all spaces with Urysohn' s f "F-property", i.e., spaces in which every closed set is a countable intersection of open sets (we call such spaces "F-spaces"). Indeed, various theorems are unnecessarily restricted to metric spaces, while weaker assertions are made about F-spaces. This seems to justify the publication of the following simple proof which extends the theorem stated above to F-spaces.

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

2016 ◽  
Vol 118 (2) ◽  
pp. 269 ◽  
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.

Densities generated by equivalent measures

2011 ◽  
Vol 61 (5) ◽  
Author(s):  
Artur Bartoszewicz ◽  
Małgorzata Filipczak ◽  
Tadeusz Poreda
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Borel Measure ◽  
The Family

AbstractIn the family of all measures equivalent to the Borel measure µ defined in a metric space X, we look for measures generating the same density points (preserving µ-density at fixed point x 0 ∈ X).

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