  Bourgain's discretization theorem asserts that there exists a universal
constant $C\in (0,\infty)$ with the following property. Let $X,Y$ be Banach
spaces with $\dim X=n$. Fix $D\in (1,\infty)$ and set $\delta= e^{-n^{Cn}}$.
Assume that $\mathcal N$ is a $\delta$-net in the unit ball of $X$ and that
$\mathcal N$ admits a bi-Lipschitz embedding into $Y$ with distortion at most
$D$. Then the entire space $X$ admits a bi-Lipschitz embedding into $Y$ with
distortion at most $CD$. This mostly expository article is devoted to a
detailed presentation of a proof of Bourgain's theorem.
  We also obtain an improvement of Bourgain's theorem in the important case
when $Y=L_p$ for some $p\in [1,\infty)$: in this case it suffices to take
$\delta= C^{-1}n^{-5/2}$ for the same conclusion to hold true. The case $p=1$
of this improved discretization result has the following consequence. For
arbitrarily large $n\in \mathbb{N}$ there exists a family $\mathscr Y$ of
$n$-point subsets of ${1,...,n}^2\subseteq \mathbb{R}^2$ such that if we write
$|\mathscr Y|= N$ then any $L_1$ embedding of $\mathscr Y$, equipped with the
Earthmover metric (a.k.a. transportation cost metric or minimumum weight
matching metric) incurs distortion at least a constant multiple of
$\sqrt{\log\log N}$; the previously best known lower bound for this problem was
a constant multiple of $\sqrt{\log\log \log N}$.
