� Measure of Cartesian Product Sets

1975 ◽  
Vol 49 (1) ◽  
pp. 199
Author(s):  
Lawrence R. Ernst
Keyword(s):  
Cartesian Product ◽  
Cartesian Product Sets ◽  
Product Sets

scholarly journals On the measure of Cartesian product sets

1950 ◽  
Vol 69 ◽  
pp. 232-232
Author(s):  
Gerald Freilich
Keyword(s):  
Cartesian Product ◽  
Cartesian Product Sets ◽  
Product Sets

An algorithm for nonlinear programs over Cartesian product sets

Optimization ◽  
1990 ◽  
Vol 21 (4) ◽  
pp. 535-542 ◽  
Author(s):  
T. Larsson ◽  
A. Migdalas
Keyword(s):  
Cartesian Product ◽  
Nonlinear Programs ◽  
Cartesian Product Sets ◽  
Product Sets

On the comparison of measure functions

1975 ◽  
Vol 78 (3) ◽  
pp. 483-491 ◽  
Author(s):  
John Hawkes
Keyword(s):  
Metric Space ◽  
Real Line ◽  
Short Proof ◽  
Cartesian Product ◽  
Cantor Set ◽  
Partial Answer ◽  
Hausdorff Measures ◽  
Cartesian Product Sets ◽  
Image Position ◽  
Product Sets

Suppose that f and g are measure functions and that µf and µg are the corresponding Hausdorff measures. We are interested in how relationships between f and g are reflected in relationships between µf and µg and vice versa. For example, ifRogers ((8), p. 80) has shown that there exists a metric space Ω which has subsets A and B with µf(A) = µg(B) = 0 and, µf(B) = µg(A) = ∞. We are more interested in what happens when the metric space in question is the real line. To get the above result we need to assume that both f and g are starshaped. This is just about a best possible result since if we assume only that f is a power of t the conclusion fails. Further-more we show that there exist measure functions f and g satisfying (1) such that µf and, µg are identical. We also consider related questions: when are, µf and, µg equivalent measures and when are they identical?The proofs depend on the construction in Theorem 3.1 of a Cantor set having pre-scribed measure properties. Although the construction is not difficult it turns out to be quite useful to appeal to the existence of such a set. We illustrate this remark by giving a short proof of a known theorem on cartesian product sets. We also make use of these ideas in section 5 where we discuss some properties of a class of net meastues and give a partial answer to a problem posed by Billingsley.

TWO MEASURE PROPERTIES OF CARTESIAN PRODUCT SETS

1954 ◽  
Vol 5 (1) ◽  
pp. 108-115 ◽  
Author(s):  
H. G. EGGLESTON
Keyword(s):  
Cartesian Product ◽  
Cartesian Product Sets ◽  
Product Sets

Packing dimension, Hausdorff dimension and Cartesian product sets

1996 ◽  
Vol 120 (3) ◽  
pp. 535-546 ◽  
Author(s):  
Yimin Xiao
Keyword(s):  
Hausdorff Dimension ◽  
Cartesian Product ◽  
Packing Dimension ◽  
Cartesian Product Sets ◽  
Product Sets

AbstractWe show that the dimension adim introduced by R. Kaufman (1987) coincides with the packing dimension Dim, but the dimension aDim introduced by Hu and Taylor [7] is different from the Hausdorff dimension. These results answer questions raised by Hu and Taylor.

A correction to a paper on the dimension of Cartesian product sets

1953 ◽  
Vol 49 (3) ◽  
pp. 437-440 ◽  
Author(s):  
H. G. Eggleston
Keyword(s):  
Hausdorff Measure ◽  
Cartesian Product ◽  
Common Point ◽  
Euclidean Spaces ◽  
Cartesian Product Sets ◽  
Image Position ◽  
Product Sets

Let Em and En be orthogonal Euclidean spaces of dimensions m and n respectively and with the origin of each as their only common point. In a previous paper (3) I gave what was intended to be a proof of the relationwhere the dimension of A, dim A, is the Besicovitch dimension, i.e. the number s such that the Hausdorff measure in any dimension greater than s is zero whilst that in any dimension less than s is infinite, where A and B are subsets of En and Em respectively and where A × B is the Cartesian product of A with B.

On Cartesian Product Sets

1952 ◽  
Vol s1-27 (3) ◽  
pp. 295-304 ◽  
Author(s):  
S. J. Taylor
Keyword(s):  
Cartesian Product ◽  
Cartesian Product Sets ◽  
Product Sets

The Besicovitch dimension of Cartesian product sets

1950 ◽  
Vol 46 (3) ◽  
pp. 383-386 ◽  
Author(s):  
H. G. Eggleston
Keyword(s):  
Euclidean Space ◽  
Hausdorff Measure ◽  
Cartesian Product ◽  
Dimensional Euclidean Space ◽  
Point Set ◽  
Cartesian Product Sets ◽  
Product Sets

A. S. Besicovitch has defined the dimension of a point-set X in n-dimensional Euclidean space in terms of its exterior Hausdorff measure as follows (2). Let (δ, X) be any enumerable class of sets whose point-set union contains X and whose members are each of diameter less than δ. Let (δ, X) denote the class of all (δ, X).

A Note on Fractal Measures and Cartesian Product Sets

2021 ◽  
Author(s):  
Najmeddine Attia ◽  
Hajer Jebali ◽  
Meriem Ben Hadj Khlifa
Keyword(s):  
Cartesian Product ◽  
Fractal Measures ◽  
Cartesian Product Sets ◽  
Product Sets

scholarly journals Notes on Hausdorff dimensions of Cartesian product sets

1971 ◽  
Vol 1 (1) ◽  
pp. 17-25 ◽  
Author(s):  
Kaoru Hatano
Keyword(s):  
Cartesian Product ◽  
Hausdorff Dimensions ◽  
Cartesian Product Sets ◽  
Product Sets
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