  We show that if $(X,\Theta)$ is a PPAV over an algebraically closed field of
characteristic $p>0$ and $D\in |m\Theta|$, then $(X,\frac 1 m D)$ is a limit of
strongly $F$-regular pairs and in particular ${\rm mult}_x(D)\leq m\cdot \dim
X$ for any $x\in X$.
