  In this paper, we generalize several results for the Hamiltonian stability
and the mean curvature flow of Lagrangian submanifolds in a K\"ahler-Einstein
manifold to more general K\"ahler manifolds including a Fano manifold equipped
with a K\"ahler form $\omega\in 2\pi c_1(M)$ by using the methodology proposed
by T. Behrndt. Namely, we first consider a weighted measure on a Lagrangian
submanifold $L$ in a K\"ahler manifold $M$ and investigate the variational
problem of $L$ for the weighted volume functional. We call a stationary point
of the weighted volume functional $f$-minimal, and define the notion of
Hamiltonian $f$-stability as a local minimizer under Hamiltonian deformations.
We show such examples naturally appear in a toric Fano manifold. Moreover, we
consider the generalized Lagrangian mean curvature flow in a Fano manifold
which is introduced by Behrndt and Smoczyk-Wang. We generalize the result of H.
Li, and show that if the initial Lagrangian submanifold is a small Hamiltonian
deformation of an $f$-minimal and Hamiltonian $f$-stable Lagrangian
submanifold, then the generalized MCF converges exponentially fast to an
$f$-minimal Lagrangian submanifold.
