Zeros of random holomorphic sections of big line bundles with continuous metrics

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 H⁰(X, L⊗ p) endowed with the inner product induced by h⊗ p and a volume form on X, and prove that the corresponding sequence of normalized Fubini-Study currents converge weakly to the curvature current c₁(L,heq) of the equilibrium metric heq 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 c₁(L,heq), for very general classes of probability measures on H⁰(X, L⊗ p).