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We develop a new approach to geometric quantization using the theory ofconvergence of metric measure spaces. Given a family of K\"ahler polarizationsconverging to a non-singular real polarization on a prequantized symplecticmanifold, we show the spectral convergence result of$\bar{\partial}$-Laplacians, as well as the convergence result of quantumHilbert spaces. We also consider the case of almost K\"ahler quantization forcompatible almost complex structures, and show the analogous convergenceresults.

Spectral convergence in geometric quantization -- the case of non-singular Langrangian fibrations