The proof of Tchakaloff’s Theorem

Source abstract

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