  We consider the metric transformation of metric measure spaces/pyramids. We
clarify the conditions to obtain the convergence of the sequence of transformed
spaces from that of the original sequence, and, conversely, to obtain the
convergence of the original sequence from that of the transformed sequence,
respectively. As an application, we prove that spheres and projective spaces
with standard Riemannian distance converge to a Gaussian space and the Hopf
quotient of a Gaussian space, respectively, as the dimension diverges to
infinity.
