Lebesgue Measure Space on the Euclidean Space

2014 ◽  
pp. 466-472
Keyword(s):  
Euclidean Space ◽  
Lebesgue Measure ◽  
Measure Space

scholarly journals Littlewood-Paley theory on Gaussian spaces

1988 ◽  
Vol 109 ◽  
pp. 47-61 ◽  
Author(s):  
Jürgen Potthoff
Keyword(s):  
Lebesgue Measure ◽  
Measure Space ◽  
Euclidean Spaces ◽  
General Measure ◽  
Finite Dimensional

In this article we prove a number of inequalities of Littlewood-Paley-Stein (LPS) type for functions on general Gaussian spaces (s. below).In finite dimensional Euclidean spaces (with Lebesgue measure) the power of such inequalities has been demonstrated in Stein’s book [12]. In his second book [13], Stein treats other spaces too: also the situation of a general measure space (X, μ). However the latter case is too general to allow for a rich class of inequalities (cf. Theorem 10 in [13]).

On the connexion between Hausdorff measures and generalized capacity

1961 ◽  
Vol 57 (3) ◽  
pp. 524-531 ◽  
Author(s):  
S. J. Taylor
Keyword(s):  
Euclidean Space ◽  
Lebesgue Measure ◽  
Hausdorff Measure ◽  
Real Function ◽  
Function Theory ◽  
Logarithmic Capacity ◽  
Hausdorff Measures ◽  
Positive Capacity ◽  
Special Case

For any real function h(t) which is continuous and monotonic increasing for t > 0 with , Hausdorff (10) in 1918 denned a Carathéodory measure with respect to h(t) which has subsequently been known as Hausdorff measure. For analysing sets in Euclidean space, these measures have proved both useful and interesting. Given a real function Φ(t) which is continuous and monotonic decreasing for t > 0 with , Frostman(9) in 1935 denned capacity with respect to Φ(t). Lebesgue measure in Euclidean k-space is a special case of Hausdorff measure, and capacity with respect to Φ(t) becomes logarithmic capacity or Newtonian capacity in the cases , Φ(t)=1/t, respectively. The interrelationship between h-measure and Φ-capacity has been of interest in both directions: (i) in applications to function theory one may be able to determine whether or not a set has positive capacity by examining the h-measure for suitable h(t) (see, for example, (5)); (ii) it may be possible to determine the measure properties of a set from knowledge of its capacity (see, for example, (7) and (17)).

The Lebesgue measure on the Euclidean space

2018 ◽  
pp. 151-185
Author(s):  
Nikos Katzourakis ◽  
Eugen Vărvărucă
Keyword(s):  
Euclidean Space ◽  
Lebesgue Measure

scholarly journals Finitely-additive invariant measures on Euclidean spaces

1982 ◽  
Vol 2 (3-4) ◽  
pp. 383-396 ◽  
Author(s):  
G. A. Margulis
Keyword(s):  
Invariant Measure ◽  
Euclidean Space ◽  
Lebesgue Measure ◽  
Invariant Measures ◽  
Dimensional Euclidean Space ◽  
Euclidean Spaces

AbstractIt is shown that for n ≥ 3 the Lebesgue measure is the unique finitely-additive isometry-invariant measure on the ring of bounded Lebesgue measurable subsets of the n-dimensional Euclidean space.

On maximal Euclidean sets avoiding certain distance configurations

1981 ◽  
Vol 89 (1) ◽  
pp. 79-88 ◽  
Author(s):  
H. T. Croft ◽  
K. J. Falconer
Keyword(s):  
Euclidean Space ◽  
Lebesgue Measure ◽  
Maximum Area

Let us say that a set points in a Euclidean space has property (2) if no 2 points X, Y ∈ S have X Y > 1. Then an easy observation is the following:The maximum area (Lebesgue measure) of a plane set S with property(2) is ¼π, attained only when S is (essentially) a disc.

scholarly journals Boundedness of Lusin-area andgλ*functions on localized Morrey-Campanato spaces over doubling metric measure spaces

2011 ◽  
Vol 9 (3) ◽  
pp. 245-282 ◽  
Author(s):  
Haibo Lin ◽  
Eiichi Nakai ◽  
Dachun Yang
Keyword(s):  
Lebesgue Measure ◽  
Measure Space ◽  
Admissible Function ◽  
Regularity Property ◽  
Metric Measure Space ◽  
Metric Measure Spaces ◽  
Area Function ◽  
Measure Spaces ◽  
Lusin Area Function ◽  
Dimensional Lebesgue Measure

Letχbe a doubling metric measure space andρan admissible function onχ. In this paper, the authors establish some equivalent characterizations for the localized Morrey-Campanato spacesερα,p(χ)and Morrey-Campanato-BLO spacesε̃ρα,p(χ)whenα∈(-∞,0)andp∈[1,∞). Ifχhas the volume regularity Property(P), the authors then establish the boundedness of the Lusin-area function, which is defined via kernels modeled on the semigroup generated by the Schrödinger operator, fromερa,p(χ)toε̃ρa,p(χ)without invoking any regularity of considered kernels. The same is true for thegλ*function and, unlike the Lusin-area function, in this case,χis even not necessary to have Property(P). These results are also new even forℝdwith thed-dimensional Lebesgue measure and have a wide applications.

scholarly journals Decision problems for linear recurrences involving arbitrary real numbers

2021 ◽  
Vol Volume 17, Issue 3 ◽  
Author(s):  
Eike Neumann
Keyword(s):  
Euclidean Space ◽  
Real Number ◽  
Lebesgue Measure ◽  
Decision Problems ◽  
Real Numbers ◽  
Linear Recurrences ◽  
Initial Values ◽  
Problem Instances ◽  
Low Order

We study the decidability of the Skolem Problem, the Positivity Problem, and the Ultimate Positivity Problem for linear recurrences with real number initial values and real number coefficients in the bit-model of real computation. We show that for each problem there exists a correct partial algorithm which halts for all problem instances for which the answer is locally constant, thus establishing that all three problems are as close to decidable as one can expect them to be in this setting. We further show that the algorithms for the Positivity Problem and the Ultimate Positivity Problem halt on almost every instance with respect to the usual Lebesgue measure on Euclidean space. In comparison, the analogous problems for exact rational or real algebraic coefficients are known to be decidable only for linear recurrences of fairly low order.

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