The volume of pseudoeffective line bundles and partial equilibrium

Let $(L,he^{-u})$ be a pseudoeffective line bundle on an $n$-dimensional compact K\"ahler manifold $X$. Let $h^0(X,L^k\otimes \mathcal I(ku))$ 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 $k^{-n}h^0(X,L^k\otimes \mathcal I(ku))$ exists, and equals the non-pluripolar volume of $P[u]_\mathcal I$, the $\mathcal I$-model potential associated to $u$. We give applications of this result to K\"ahler quantization: fixing a Bernstein-Markov measure $\nu$, we show that the partial Bergman measures of $u$ converge weakly to the non-pluripolar Monge--Amp\`ere measure of $P[u]_\mathcal I$, the partial equilibrium.