Let T be a measure-preserving ergodic transformation of a measure space (X,,μ) and, for f ∈ L(X), let $f*=su{p}_{N}1/N{\sum }_{m=0}^{N-1}f\circ T^m$. In this paper we mainly investigate the question of whether (i) ${ʃ}_a^{\infty }|\mu (f*>t)-1/t{ʃ}_{(f*>t)}fd\mu |dt<\infty$ and whether (ii) ${ʃ}_a^{\infty }|\mu (f*>t)-1/t{ʃ}_{(f>t)}fd\mu |dt<\infty$ for some a > 0. It is proved that (i) holds for every f ≥ 0. (ii) holds if f ≥ 0 and f log log (f + 3) ∈ L(X) or if μ(X) = 1 and the random variables $f\circ T^m$ are independent. Related inequalities are proved. Some examples and counterexamples are constructed. Several known results are obtained as corollaries.

Let (X,,μ,τ) be an ergodic dynamical system and φ be a measurable map from X to a locally compact second countable group G with left Haar measure $m_G$. We consider the map ${\tau }_{\phi }$ defined on X × G by ${\tau }_{\phi }:(x,g)\mapsto (\tau x,\phi \left(x\right)g)$ and the cocycle ${\left(\phi ₙ\right)}_{n\in \mathbb{Z}}$ generated by φ. Using a characterization of the ergodic invariant measures for ${\tau }_{\phi }$, we give the form of the ergodic decomposition of $\mu \left(dx\right)\otimes m_G\left(dg\right)$ or more generally of the ${\tau }_{\phi }$-invariant measures ${\mu }_{\chi }\left(dx\right)\otimes \chi \left(g\right)m_G\left(dg\right)$, where ${\mu }_{\chi }\left(dx\right)$ is χ∘φ-conformal for an exponential χ on G.

The Stein-Weiss theorem that the distribution function of the Hilbert transform of the characteristic function of E depends only on the measure of E is generalized to the ergodic Hilbert transform.

A generalization of the Avez method of construction of an invariant measure is presented.

For an arbitrary set $A\subseteq {\mathbb{N}}^{+}$ containing 1, an ergodic automorphism T whose set of essential values of the multiplicity function is equal to A is constructed. If A is additionally finite, T can be chosen to be an analytic diffeomorphism on a finite-dimensional torus.

For an arbitrary set A ⊆ ℕ satisfying 1 ∈ A and lcm(m₁,m₂) ∈ A whenever m₁,m₂ ∈ A, an ergodic abelian group extension of a rotation for which the range of the multiplicity function equals A is constructed.