Results 151 to 160 of about 2,217 (207)
Some of the next articles are maybe not open access.
2020
It was mentioned that to set up a theory of measure capable of measuring the greatest number of subsets in a given space, Lebesgue started with their outer measure \(m^*\), dropped the property of invariance under complements and as counterpart defined inner measures in terms of outer measures of complements.
openaire +1 more source
It was mentioned that to set up a theory of measure capable of measuring the greatest number of subsets in a given space, Lebesgue started with their outer measure \(m^*\), dropped the property of invariance under complements and as counterpart defined inner measures in terms of outer measures of complements.
openaire +1 more source
2013
This chapter is devoted to the very important notion of Lebesbue measure in a finite-dimensional Euclidean space. We give a classical example of an uncountable linear set of Lebesgue measure zero (the Cantor set), and also an example of a Lebesgue non-measurable set. In Sects. 2.2 and 2.3, we establish deeper properties of the Lebesgue measure, such as
Boris Makarov, Anatolii Podkorytov
openaire +1 more source
This chapter is devoted to the very important notion of Lebesbue measure in a finite-dimensional Euclidean space. We give a classical example of an uncountable linear set of Lebesgue measure zero (the Cantor set), and also an example of a Lebesgue non-measurable set. In Sects. 2.2 and 2.3, we establish deeper properties of the Lebesgue measure, such as
Boris Makarov, Anatolii Podkorytov
openaire +1 more source
Slice Lebesgue Measure of Quaternions
Advances in Applied Clifford Algebras, 2015zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Ren, Guangbin, Xu, Zhenghua
openaire +1 more source
2019
It is shown that the notion of the length of an interval defines a measure on the ring generated by left-closed-right-open intervals in the real line ℝ. Using the method of Caratheodory, the Lebesgue measure is constructed on ℝ and its important properties are studied. Generalizations to ℝN,N ≥ 2, are also described.
openaire +1 more source
It is shown that the notion of the length of an interval defines a measure on the ring generated by left-closed-right-open intervals in the real line ℝ. Using the method of Caratheodory, the Lebesgue measure is constructed on ℝ and its important properties are studied. Generalizations to ℝN,N ≥ 2, are also described.
openaire +1 more source
Bessel-Riesz Operators on Lebesgue Spaces with Lebesgue Measures
Malaysian Journal of Mathematical SciencesThis study investigates a class of mathematical operators known as the Bessel-Riesz operators, defined in Euclidean space Rn, given by, Tμ,νf(z) =Z Rn Kμ,ν (|z − w|)f(w)dν(w), for z ∈ Rn. (1) Here, Kμ,ν is called the Bessel-Riesz kernel. It can be expressed as a multiple of the Bessel kernel Jν and the Riesz kernel Kμ.
S. Mehmood +3 more
openaire +1 more source
The Mathematical Intelligencer, 1989
A beautiful expository paper on positive and negative results concerning the extension of Lebesgue measure in \(R^ n\) and their relation to set theoretical hypotheses. (Reviewer's remark: The Tarski circle-squaring problem, mentioned as an open one, has been solved meanwhile by \textit{M. Laczkovich} [J. Reine Angew. Math. 404,77-117 (1990)].)
openaire +2 more sources
A beautiful expository paper on positive and negative results concerning the extension of Lebesgue measure in \(R^ n\) and their relation to set theoretical hypotheses. (Reviewer's remark: The Tarski circle-squaring problem, mentioned as an open one, has been solved meanwhile by \textit{M. Laczkovich} [J. Reine Angew. Math. 404,77-117 (1990)].)
openaire +2 more sources
On the Uniqueness of Lebesgue and Borel Measures
Journal of Applied Analysis, 1997Summary: We consider the uniqueness property for various invariant measures. Primarily, we discuss this property for the standard Lebesgue measure on the \(n\)-dimensional Euclidean space \(\mathbb{R}^n\) (sphere \(\mathbb{S}^n\)) and for the standard Borel measure on the same space (sphere), which is the restriction of the Lebesgue measure to the ...
openaire +2 more sources
LOCATEDNESS, CONVEXITY, AND LEBESGUE MEASURABILITY
The Quarterly Journal of Mathematics, 1988It cannot be shown constructively that every compact set in the Euclidean space is Lebesgue measurable. This paper studies the constructive measurability of convex sets. The concept of locatedness, ubiquitous in constructive analysis, is central to the results.
openaire +2 more sources
Continuous transforms of a Lebesgue measure
Mathematical Notes, 1999The following statement is proved: Theorem 1. Let \(K\) be a compact metric space which is the continuous image of \([0,1]\) and \(\mu\) be a Borel probability measure on \(K\) whose topological support coincides with \(K\). Then there exists a continuous surjective \(g:[0,1]\to K\) such that \(\lambda\circ g^{-1}=\mu,\) where \(\lambda\) is the ...
openaire +2 more sources