  We develop a new approach to geometric quantization using the theory of
convergence of metric measure spaces. Given a family of K\"ahler polarizations
converging to a non-singular real polarization on a prequantized symplectic
manifold, we show the spectral convergence result of
$\bar{\partial}$-Laplacians, as well as the convergence result of quantum
Hilbert spaces. We also consider the case of almost K\"ahler quantization for
compatible almost complex structures, and show the analogous convergence
results.
