The fractal dimension of invariant subsets for piecewise monotonic maps on the interval.
The generalized purity law for ergodic measures: a simple proof
The quasi-sure ratio ergodic theorem
The σ-ideal of closed smooth sets does not have the covering property
We prove that the σ-ideal I(E) (of closed smooth sets with respect to a non-smooth Borel equivalence relation E) does not have the covering property. In fact, the same holds for any σ-ideal containing the closed transversals with respect to an equivalence relation generated by a countable group of homeomorphisms. As a consequence we show that I(E) does not have a Borel basis.