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Let $\Gamma$ be a countably infinite group. A common theme in ergodic theoryis 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 theglobal 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 anonempty finite subset of $\Gamma$. The basic hope is that, when $D$ runs overa suitably chosen infinite sequence, these pointwise averages should convergeto the global value for $\mu$-almost all $x$.  In this paper we prove several results that refine the above basic paradigmby uniformly controlling the averages over specific sets $D$ rather thanconsidering their limit as $|D| \to \infty$. Our results include ergodictheorems for the Bernoulli shift action $\Gamma \curvearrowright ([0;1]^\Gamma,\lambda^\Gamma)$ and strengthenings of the theorem of Ab\'ert and Weiss thatthe shift is weakly contained in every free p.m.p. action of $\Gamma$. Inparticular, we establish a purely Borel version of the Ab\'ert--Weiss theoremfor finitely generated groups of subexponential growth. The central role in ourarguments is played by the recently introduced measurable versions of theLov\'asz Local Lemma, due to the current author and to Cs\'oka, Grabowski,M\'ath\'e, Pikhurko, and Tyros.

Ergodic Theorems for the Shift Action and Pointwise Versions of The Ab\'ert--Weiss Theorem