  In this article, we consider minimal $L^2$ integrals on the sublevel sets of
plurisubharmonic functions on weakly pseudoconvex K\"ahler manifolds with
Lebesgue measurable gain related to modules at boundary points of the sublevel
sets, and establish a concavity property of the minimal $L^2$ integrals. As
applications, we present a necessary condition for the concavity degenerating
to linearity, a concavity property related to modules at inner points of the
sublevel sets, an optimal support function related to the modules, a strong
openness property of the modules and a twisted version, an effectiveness result
of the strong openness property of the modules.
