Concavity property of minimal $L^2$ integrals with Lebesgue measurable gain II

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 Kähler 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.