  For a given metric measure space $(X,d,\mu)$ we consider finite samples of
points, calculate the matrix of distances between them and then reconstruct the
points in some finite-dimensional space using the multidimensional scaling
(MDS) algorithm with this distance matrix as an input. We show that this
procedure gives a natural limit as the number of points in the samples grows to
infinity and the density of points approaches the measure $\mu$. This limit can
be viewed as "infinite MDS" embedding of the original space, now not anymore
into a finite-dimensional space but rather into an infinitedimensional Hilbert
space. We further show that this embedding is stable with respect to the
natural convergence of metric measure spaces. However, contrary to what is
usually believed in applications, we show that in many cases it does not
preserve distances, nor is even bi-Lipschitz, but may provide snowflake
(Assouad-type) embeddings of the original space to a Hilbert space (this is,
for instance, the case of a sphere and a flat torus equipped with their
geodesic distances).
