Weakly mixing transformations and the Carathéodory definition of measurable sets
Volume 108 / 2007
Abstract
Let ${\Bbb T}$ 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 ${\Bbb T}$.
