Hopf's ratio ergodic theorem by inducing
We present a very quick and easy proof of the classical Stepanov-Hopf ratio ergodic theorem, deriving it from Birkhoff's ergodic theorem by a simple inducing argument.
Hopf's theorem on invariant measures for a group of transformations
How many points contain arithmetic progressions in their continued fraction expansion?
How smooth is almost every function in a Sobolev space?
We show that almost every function (in the sense of prevalence) in a Sobolev space is multifractal: Its regularity changes from point to point; the sets of points with a given Hölder regularity are fractal sets, and we determine their Hausdorff dimension.
How subadditive are subadditive capacities?
Subadditivity of capacities is defined initially on the compact sets and need not extend to all sets. This paper explores to what extent subadditivity holds. It presents some incidental results that are valid for all subadditive capacities. The main result states that for all hull-additive capacities (a class that contains the strongly subadditive capacities) there is countable subadditivity on a class at least as large as the universally measurable sets (so larger than the analytic sets).
Hrášek a sluníčko. O matematickém paradoxu Stefana Banacha a Alfreda Tarského
H-regularidad para medidas vectoriales.
Hyper-extensions of -algebras