The volume of pseudoeffective line bundles and partial equilibrium

Let (L, he(-u)) be a pseudoeffective line bundle on an n-dimensional compact Kahler manifold X. Let h(0) (X, L-k circle times I(ku)) be the dimension of the space of sections s of L-k such that h(k) (s, s)e(-ku) is integrable. We show that the limit of k(-n)h(0)(X, L-k circle times I(ku)) exists, and equals the nonpluripolar volume of P[u](I), the I-model potential associated to u. We give applications of this result to Kahler quantization: fixing a Bernstein-Markov measure nu, we show that the partial Bergman measures of u converge weakly to the nonpluripolar Monge-Ampere measure of P[u](I), the partial equilibrium.