  We study the Borel complexity of sets of normal numbers in several numeration
systems. Taking a dynamical point of view, we offer a unified treatment for
continued fraction expansions and base $r$ expansions, and their various
generalisations: generalised L\"uroth series expansions and $\beta$-expansions.
In fact, we consider subshifts over a countable alphabet generated by all
possible expansions of numbers in $[0,1)$. Then normal numbers correspond to
generic points of shift-invariant measures. It turns out that for these
subshifts the set of generic points for a shift-invariant probability measure
is precisely at the third level of the Borel hierarchy (it is a
$\Pi^0_3$-complete set, meaning that it is a countable intersection of
$F_\sigma$-sets, but it is not possible to write it as a countable union of
$G_\delta$-sets). We also solve a problem of Sharkovsky--Sivak on the Borel
complexity of the basin of statistical attraction. The crucial dynamical
feature we need is a feeble form of specification. All expansions named above
generate subshifts with this property. Hence the sets of normal numbers under
consideration are $\Pi^0_3$-complete.
