A theorem of Kučera states that given a Martin-Löf random infinite binary sequence ω and an effectively open set A of measure less than 1, some tail of ω is not in A. We show that this result can be seen as an effective version of Birkhoff’s ergodic theorem (in a special case). We prove several results in the same spirit and generalize them via an effective ergodic theorem for bijective ergodic...

The uniqueness theorem for the ergodic maximal operator is proved in the continuous case. Let (X, S, μ) be a finite measure space, μ(X) < ∞, (1) and let (Tt)t≥0 be an ergodic semigroup of measure-preserving transformations of (X, S, μ). As usual the map (x, t) → Ttx is assumed to be jointly measurable. For an integrable function f , f ∈ L(X), the ergodic maximal function f∗ is defined by equati...

An open problem in polarization theory is to determine the binary operations that always lead to polarization (in the general multilevel sense) when they are used in Arıkan style constructions. This paper, which is presented in two parts, solves this problem by providing a necessary and sufficient condition for a binary operation to be polarizing. This (first) part of the paper introduces the m...