# Weakly mixing transformations and the Carathéodory definition of measurable sets

Colloquium Mathematicae (2007)

• Volume: 108, Issue: 2, page 317-328
• ISSN: 0010-1354

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## Abstract

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Let 𝕋 denote the set of complex numbers of modulus 1. Let v ∈ 𝕋, v not a root of unity, and let T: 𝕋 → 𝕋 be the transformation on 𝕋 given by T(z) = vz. It is known that the problem of calculating the outer measure of a T-invariant set leads to a condition which formally has a close resemblance to Carathéodory's definition of a measurable set. In ergodic theory terms, T is not weakly mixing. Now there is an example, due to Kakutani, of a transformation ψ̃ which is weakly mixing but not strongly mixing. The results here show that the problem of calculating the outer measure of a ψ̃-invariant set leads to a condition formally resembling the Carathéodory definition, as in the case of the transformation T. The methods used bring out some of the more detailed behaviour of the Kakutani transformation. The above mentioned results for T and the Kakutani transformation do not apply for the strongly mixing transformation z ↦ z² on 𝕋.

## How to cite

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Amos Koeller, Rodney Nillsen, and Graham Williams. "Weakly mixing transformations and the Carathéodory definition of measurable sets." Colloquium Mathematicae 108.2 (2007): 317-328. <http://eudml.org/doc/286179>.

@article{AmosKoeller2007,
abstract = {Let 𝕋 denote the set of complex numbers of modulus 1. Let v ∈ 𝕋, v not a root of unity, and let T: 𝕋 → 𝕋 be the transformation on 𝕋 given by T(z) = vz. It is known that the problem of calculating the outer measure of a T-invariant set leads to a condition which formally has a close resemblance to Carathéodory's definition of a measurable set. In ergodic theory terms, T is not weakly mixing. Now there is an example, due to Kakutani, of a transformation ψ̃ which is weakly mixing but not strongly mixing. The results here show that the problem of calculating the outer measure of a ψ̃-invariant set leads to a condition formally resembling the Carathéodory definition, as in the case of the transformation T. The methods used bring out some of the more detailed behaviour of the Kakutani transformation. The above mentioned results for T and the Kakutani transformation do not apply for the strongly mixing transformation z ↦ z² on 𝕋.},
author = {Amos Koeller, Rodney Nillsen, Graham Williams},
journal = {Colloquium Mathematicae},
keywords = {ergodic transformation; mixing; outer measure; Kakutani transformation; measurable set; Carathéodory definition},
language = {eng},
number = {2},
pages = {317-328},
title = {Weakly mixing transformations and the Carathéodory definition of measurable sets},
url = {http://eudml.org/doc/286179},
volume = {108},
year = {2007},
}

TY - JOUR
AU - Amos Koeller
AU - Rodney Nillsen
AU - Graham Williams
TI - Weakly mixing transformations and the Carathéodory definition of measurable sets
JO - Colloquium Mathematicae
PY - 2007
VL - 108
IS - 2
SP - 317
EP - 328
AB - Let 𝕋 denote the set of complex numbers of modulus 1. Let v ∈ 𝕋, v not a root of unity, and let T: 𝕋 → 𝕋 be the transformation on 𝕋 given by T(z) = vz. It is known that the problem of calculating the outer measure of a T-invariant set leads to a condition which formally has a close resemblance to Carathéodory's definition of a measurable set. In ergodic theory terms, T is not weakly mixing. Now there is an example, due to Kakutani, of a transformation ψ̃ which is weakly mixing but not strongly mixing. The results here show that the problem of calculating the outer measure of a ψ̃-invariant set leads to a condition formally resembling the Carathéodory definition, as in the case of the transformation T. The methods used bring out some of the more detailed behaviour of the Kakutani transformation. The above mentioned results for T and the Kakutani transformation do not apply for the strongly mixing transformation z ↦ z² on 𝕋.
LA - eng
KW - ergodic transformation; mixing; outer measure; Kakutani transformation; measurable set; Carathéodory definition
UR - http://eudml.org/doc/286179
ER -

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