  Let $\Gamma$ be a countably infinite group. A common theme in ergodic theory
is to start with a probability measure-preserving (p.m.p.) action $\Gamma
\curvearrowright (X, \mu)$ and a map $f \in L^1(X, \mu)$, and to compare the
global average $\int f \,\mathrm{d}\mu$ of $f$ to the pointwise averages
$|D|^{-1} \sum_{\delta \in D} f(\delta \cdot x)$, where $x \in X$ and $D$ is a
nonempty finite subset of $\Gamma$. The basic hope is that, when $D$ runs over
a suitably chosen infinite sequence, these pointwise averages should converge
to the global value for $\mu$-almost all $x$.
  In this paper we prove several results that refine the above basic paradigm
by uniformly controlling the averages over specific sets $D$ rather than
considering their limit as $|D| \to \infty$. Our results include ergodic
theorems for the Bernoulli shift action $\Gamma \curvearrowright ([0;1]^\Gamma,
\lambda^\Gamma)$ and strengthenings of the theorem of Ab\'ert and Weiss that
the shift is weakly contained in every free p.m.p. action of $\Gamma$. In
particular, we establish a purely Borel version of the Ab\'ert--Weiss theorem
for finitely generated groups of subexponential growth. The central role in our
arguments is played by the recently introduced measurable versions of the
Lov\'asz Local Lemma, due to the current author and to Cs\'oka, Grabowski,
M\'ath\'e, Pikhurko, and Tyros.
