  We propose a definition of the weighted $\sigma_k$-curvature of a smooth
metric measure space and justify it in two ways. First, we show that the
weighted $\sigma_k$-curvature prescription problem is governed by a fully
nonlinear second order elliptic PDE which is variational when $k=1,2$ or the
smooth metric measure space is locally conformally flat in the weighted sense.
Second, we show that, in the variational cases, quasi-Einstein metrics are
stable with respect to the total weighted $\sigma_k$-curvature functional. We
also discuss related conjectures for weighted Einstein manifolds.
