  Tchakaloff's Theorem establishes the existence of a quadrature rule of
prescribed degree relative to a positive, compactly supported measure that is
absolutely continuous with respect to Lebesgue measure on $\mathbb{R}^{d}$.
Subsequent extensions were obtained by Mysovskikh and by Putinar. We provide
new proofs and partial extensions of these results, based on duality techniques
utilized by Stochel. We also obtain new uniqueness criteria in the Truncated
Complex Moment Problem.
