Weakly mixing transformations and the Carathéodory definition of measurable sets
Volume 108 / 2007
Abstract
Let denote the set of complex numbers of modulus 1. Let v\in{\Bbb T}, v not a root of unity, and let T:{\Bbb T}\rightarrow {\Bbb T} be the transformation on {\Bbb 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 \widetilde \psi which is weakly mixing but not strongly mixing. The results here show that the problem of calculating the outer measure of a \widetilde \psi-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\mapsto z^2 on {\Bbb T}.
