  Let $(X,\omega)$ be a compact K\"ahler manifold of dimension $n$ and fix
$m\in \mathbb{N}$ such that $1\leq m \leq n$. We prove that any $(\omega,m)$-sh
function can be approximated from above by smooth $(\omega,m)$-sh functions. A
potential theory for the complex Hessian equation is also developed which
generalizes the classical pluripotential theory on compact K\"ahler manifolds.
We then use novel variational tools due to Berman, Boucksom, Guedj and Zeriahi
to study degenerate complex Hessian equations.
