  The discrete data encoded in the power moments of a positive measure, fast
decaying at infinity on euclidean space, is incomplete for recovery, leading to
the concept of moment indeterminateness. On the other hand, classical integral
transforms (Fourier-Laplace, Fantappi\`e, Poisson) of such measures are
complete, often invertible via an effective inverse operation. The gap between
the two non-uniqueness/ uniqueness phenomena is manifest in the dual picture,
when trying to extend the measure, regarded as a positive linear functional,
from the polynomial algebra to the full space of continuous functions. This
point of view was advocated by Marcel Riesz a century ago, in the single real
variable setting. Notable advances in functional analysis have root in Riesz'
celebrated four notes devoted to the moment problem. A key technical ingredient
being there the monotone approximation by polynomials of kernels of integral
transforms. With inherent new obstacles we reappraise in the context of several
real variables M. Riesz' variational principle. The result is an array of
necessary and sufficient moment indeterminateness criteria, some raising real
algebra questions, others involving intriguing analytic problems, all
gravitating around the concept of moment separating function.
