In this paper we investigate the extent to which the Lovász Local Lemma (an important tool in probabilistic combinatorics) can be adapted for the measurable setting. In most applications, the Lovász Local Lemma is used to produce a function $f \colon X \to Y$ with certain properties, where $X$ is some underlying combinatorial structure and $Y$ is a (typically finite) set. Can this function $f$ be chosen to be Borel or $μ$-measurable for some probability Borel measure $μ$ on $X$ (assuming that $X$ is a standard Borel space)? In the positive direction, we prove that if the set of constraints put on $f$ is, in a certain sense, "locally finite," then there is always a Borel choice for $f$ that is "$\varepsilon$-close" to satisfying these constraints, for any $\varepsilon > 0$. Moreover, if the combinatorial structure on $X$ is "induced" by the $[0;1]$-shift action of a countable group $Γ$, then, even without any local finiteness assumptions, there is a Borel choice for $f$ which satisfies the constraints on an invariant conull set (i.e., with $\varepsilon = 0$). A direct corollary of our results is an upper bound on the measurable chromatic number of the graph $G_n$ generated by the shift action of the free group $\mathbb{F}_n$ that is asymptotically tight up to a factor of at most $2$ (which answers a question of Lyons and Nazarov). On the other hand, our result for structures induced by measure-preserving group actions is, at least for amenable groups, sharp in the following sense: a probability measure-preserving action of a countably infinite amenable group satisfies the measurable version of the Lovász Local Lemma if and only if it admits a factor map to the $[0;1]$-shift action. To prove this, we combine the tools of the Ornstein--Weiss theory of entropy for actions of amenable groups with concepts from computability theory, specifically, Kolmogorov complexity.