  In this article, we present a concavity property of the minimal $L^{2}$
integrals related to multiplier ideal sheaves with Lebesgue measurable gain on
weakly pseudoconvex K\"ahler manifolds. As applications, we give a necessary
condition for the concavity degenerating to linearity, and a characterization
for the holding of the equality in optimal jets $L^2$ extension problem on open
Riemann surfaces.
