The proof of Tchakaloff's theorem

We provide a simple proof of Tchakalo.'s Theorem on the existence of cubature formulas of degree m for Borel measures with moments up to order m. The result improves known results for non-compact support, since we do not need conditions on (m+1) st moments. In fact, we reduce the classical assertion of Tchakalo.'s Theorem to a well-known statement going back to F. Riesz.