PomeCorollary 18.34.2.E8DF257514D4 If $N_ m=o(m^ n)$, then $\beta _ m^{(n)}\to 0$ in total variation, and $\mathcal F_ m^{(n)}(v)\to 0$ uniformly for $v$ in any bounded subset of $C(X)$.
Corollary 18.34.2E8DF257514D4
Proof. The measures $\beta _ m^{(n)}$ are positive with total masses $n!N_ m/m^ n\to 0$. The second assertion follows from the Lipschitz estimate, since $\mathcal F_ m^{(n)}(0)=0$. $\square$
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