  This paper is concerned with the study of the consistency of a variational
method for probability measure quantization, deterministically realized by
means of a minimizing principle, balancing power repulsion and attraction
potentials. The proof of consistency is based on the construction of a target
energy functional whose unique minimizer is actually the given probability
measure \omega to be quantized. Then we show that the discrete functionals,
defining the discrete quantizers as their minimizers, actually \Gamma-converge
to the target energy with respect to the narrow topology on the space of
probability measures. A key ingredient is the reformulation of the target
functional by means of a Fourier representation, which extends the
characterization of conditionally positive semi-definite functions from points
in generic position to probability measures. As a byproduct of the Fourier
representation, we also obtain compactness of sublevels of the target energy in
terms of uniform moment bounds, which already found applications in the
asymptotic analysis of corresponding gradient flows. To model situations where
the given probability is affected by noise, we additionally consider a modified
energy, with the addition of a regularizing total variation term and we
investigate again its point mass approximations in terms of \Gamma-convergence.
We show that such a discrete measure representation of the total variation can
be interpreted as an additional nonlinear potential, repulsive at a short
range, attractive at a medium range, and at a long range not having effect,
promoting a uniform distribution of the point masses.
