  Let $(X,\omega)$ be a compact K\"ahler manifold of dimension $n$, and
$\theta$ be a closed smooth real $(1,1)$-form representing a big and nef
cohomology class. We introduce a metric $d_p, p\geq 1$, on the finite energy
space $\mathcal{E}^p(X,\theta)$, making it a complete geodesic metric space.
