Stretched shadings and a Banach measure that is not scale-invariant

Richard D. Mabry. "Stretched shadings and a Banach measure that is not scale-invariant." Fundamenta Mathematicae 209.2 (2010): 95-113. <http://eudml.org/doc/283391>.

@article{RichardD2010,
abstract = {It is shown that if A ⊂ ℝ has the same constant shade with respect to all Banach measures, then the same is true of any similarity transformation of A and the shade is not changed by the transformation. On the other hand, if A ⊂ ℝ has constant μ-shade with respect to some fixed Banach measure μ, then the same need not be true of a similarity transformation of A with respect to μ. But even if it is, the μ-shade might be changed by the transformation. To prove such a μ exists, a Hamel basis with some weak closure properties with respect to multiplication is used to construct sets with some convenient scaling properties. The notion of shade-almost invariance is introduced, aiding in the construction of a variety of Banach measures, in particular, one that is not scale-invariant.},
author = {Richard D. Mabry},
journal = {Fundamenta Mathematicae},
keywords = {Banach measure; Lebesgue measure; scale-invariance; Hamel basis; shading; shade-almost invariance; invariant ideal},
language = {eng},
number = {2},
pages = {95-113},
title = {Stretched shadings and a Banach measure that is not scale-invariant},
url = {http://eudml.org/doc/283391},
volume = {209},
year = {2010},
}

TY - JOUR
AU - Richard D. Mabry
TI - Stretched shadings and a Banach measure that is not scale-invariant
JO - Fundamenta Mathematicae
PY - 2010
VL - 209
IS - 2
SP - 95
EP - 113
AB - It is shown that if A ⊂ ℝ has the same constant shade with respect to all Banach measures, then the same is true of any similarity transformation of A and the shade is not changed by the transformation. On the other hand, if A ⊂ ℝ has constant μ-shade with respect to some fixed Banach measure μ, then the same need not be true of a similarity transformation of A with respect to μ. But even if it is, the μ-shade might be changed by the transformation. To prove such a μ exists, a Hamel basis with some weak closure properties with respect to multiplication is used to construct sets with some convenient scaling properties. The notion of shade-almost invariance is introduced, aiding in the construction of a variety of Banach measures, in particular, one that is not scale-invariant.
LA - eng
KW - Banach measure; Lebesgue measure; scale-invariance; Hamel basis; shading; shade-almost invariance; invariant ideal
UR - http://eudml.org/doc/283391
ER -