  Hajnal and Szemer\'{e}di proved that if $G$ is a finite graph with maximum
degree $\Delta$, then for every integer $k \geqslant \Delta+1$, $G$ has a
proper coloring with $k$ colors in which every two color classes differ in size
at most by $1$; such colorings are called equitable. We obtain an analog of
this result for infinite graphs in the Borel setting. Specifically, we show
that if $G$ is an aperiodic Borel graph of finite maximum degree $\Delta$, then
for each $k \geqslant \Delta + 1$, $G$ has a Borel proper $k$-coloring in which
every two color classes are related by an element of the Borel full semigroup
of $G$. In particular, such colorings are equitable with respect to every
$G$-invariant probability measure. We also establish a measurable version of a
result of Kostochka and Nakprasit on equitable $\Delta$-colorings of graphs
with small average degree. Namely, we prove that if $\Delta \geqslant 3$, $G$
does not contain a clique on $\Delta + 1$ vertices, and $\mu$ is an atomless
$G$-invariant probability measure such that the average degree of $G$ with
respect to $\mu$ is at most $\Delta/5$, then $G$ has a $\mu$-equitable
$\Delta$-coloring. As steps towards the proof of this result, we establish
measurable and list coloring extensions of a strengthening of Brooks's theorem
due to Kostochka and Nakprasit.
