  Generalizing classic results for a family of measures in the torus, for a
family $(\mu_t)_{t\geq 0}$ of measures defined on a nilmanifold $X$, we study
conditions under which the family equidistributes, meaning conditions under
which the measures $\mu_t$ converge as $t\to\infty$ in the weak$^\ast$ topology
to the Haar measure on $X$. We give general conditions on a family of measures
defined by a dilation process, showing necessary and sufficient conditions for
equidistribution as the family dilates, along with conditions such that this
holds for all dilates outside some set of density zero. Furthermore, we show
that these two types of equidistribution are different.
