  Let $\mathcal G$ be a family of subsets of an $n$-element set. The family
$\mathcal G$ is called non-trivial $3$-wise intersecting if the intersection of
any three subsets in $\mathcal G$ is non-empty, but the intersection of all
subsets is empty. For a real number $p\in(0,1)$ we define the measure of the
family by the sum of $p^{|G|}(1-p)^{n-|G|}$ over all $G\in\mathcal G$. We
determine the maximum measure of non-trivial $3$-wise intersecting families. We
also discuss the uniqueness and stability of the corresponding optimal
structure. These results are obtained by solving linear programming problems.
