  Two families, ${\mathcal A}$ and ${\mathcal B}$, of subsets of $[n]$ are
cross $t$-intersecting if for every $A \in {\mathcal A}$ and $B \in {\mathcal
B}$, $A$ and $B$ intersect in at least $t$ elements. For a real number $p$ and
a family ${\mathcal A}$ the product measure $\mu_p ({\mathcal A})$ is defined
as the sum of $p^{|A|}(1-p)^{n-|A|}$ over all $A\in{\mathcal A}$. For every
non-negative integer $r$, and for large enough $t$, we determine, for any $p$
satisfying $\frac r{t+2r-1}\leq p\leq\frac{r+1}{t+2r+1}$, the maximum possible
value of $\mu_p ({\mathcal A})\mu_p ({\mathcal B})$ for cross $t$-intersecting
families ${\mathcal A}$ and ${\mathcal B}$. In this paper we prove a stronger
stability result which yields the above result.
