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

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Mabry R.

Fundamenta Mathematicae

2010

tom 209

nr 2

95-113

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.

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1 Fundamenta Mathematicae

1 Mabry R.

1 2010

Wyniki wyszukiwania

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