Let $X$ be a compact normal complex space, $L$ be a big holomorphic line bundle on $X$ and $h$ be a continuous Hermitian metric on $L$. We consider the spaces of holomorphic sections $H^0(X, L^{\otimes p})$ endowed with the inner product induced by $h^{\otimes p}$ and a volume form on $X$, and prove that the corresponding sequence of normalized Fubini-Study currents converge weakly to the curvature current $c_1(L,h_{\mathrm{eq}})$ of the equilibrium metric $h_{\mathrm{eq}}$ associated to $h$. We also show that the normalized currents of integration along the zero divisors of random sequences of holomorphic sections converge almost surely to $c_1(L,h_{\mathrm{eq}})$, for very general classes of probability measures on $H^0(X, L^{\otimes p})$.