  Given a $\mathbb Z^r$-action $\alpha$ on a nilmanifold $X$ by automorphisms
and an ergodic $\alpha$-invariant probability measure $\mu$, we show that $\mu$
is the uniform measure on $X$, unless modulo finite index modification, one of
the following obstructions occurs for an algebraic factor action: 1. The factor
measure has zero entropy under every element of the action; 2. The factor
action is virtually cyclic. We also deduce a rigidity property for invariant
closed subsets.
