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Let $L$ be a holomorphic line bundle on a compact complex manifold $X$ of dimension $n,$ and let $e^{-\phi}$ be a continuous metric on $L.$ Fixing a measure $d\mu$ on $X$ gives a sequence of Hilbert spaces consisting of holomorphic sections of tensor powers of $L.$ We prove that the corresponding sequence of scaled Bergman measures converges, in the high tensor power limit, to the equilibrium measure of the pair $(K,\phi),$ where $K$ is the support of $d\mu,$ as long as $d\mu$ is stably Bernstein-Markov with respect to $(K,\phi).$ Here the Bergman measure denotes $d\mu$ times the restriction to the diagonal of the pointwise norm of the corresponding orthogonal projection operator.
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R Berman Bergman Kernels and Equilibrium Measures for Line Bundles over Projective Manifolds Preprint in 2007 at arXiv.org, arXiv:math/0710.4375v2
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T Bloom and N Levenberg Strong Asymptotics for Christoffel Functions of Planar Measures Preprint in 2007 at arXiv.org, arXiv:math/0709.2073v1
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R Berman Bergman Kernels and Equilibrium Measures for Polarized Pseudoconcave Domains Preprint in 2008 at arXiv.org, arXiv:math/0608226v3
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R Berman Bergman Kernels for Weighted Polynomials and Weighted Equilibrium Measures of ℂ n \mathbb{C}^{n} Preprint in 2008 at arXiv.org, arXiv:math/0702357v2
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R Berman and S Boucksom Capacities and Weighted Volumes of Line Bundles Preprint in 2008 at arXiv.org, arXiv:math/0803.1950v1
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2007
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