  In this article, we present a concavity property of the minimal $L^2$
integrals related to multiplier ideal sheaves with Lebesgue measurable gain. As
applications, we give necessary conditions for our concavity degenerating to
linearity, characterizations for 1-dimensional case, and a characterization for
the holding of the equality in optimal $L^2$ extension problem on open Riemann
surfaces with weights may not be subharmonic.
