The following is a standard, but I couldn't find a single textbook for it. Could you help me find a reference for this lemma?

Given $f,f_N \in L^2 (\Omega) \cap L^1 (\Omega)$ for some probability space $(\Omega , \mu )$, then $$\left| \mathbb{E}_{\alpha} [f] - \mathbb{E}_{\alpha} [f_N] \right| \leq \|f - f_N\|_2 \, ,$$ $$ \left| {\rm Var}(f) - {\rm Var}(f_N) \right| \leq \|f-f_N\|_2 \cdot (\|f\|_2 +\|f_N\|_2 ) \, , $$ $$ |\sigma(f)-\sigma(f_N) | \leq \frac{\|f\|_2 + \|f_N\|_2}{\sigma(f) +\sigma(f_N)} \|f-f_N\|_2 \, . $$

In other words, the $L^2$ error bounds the moment approximation error.

Note- This is somewhat overlapping with this MathOverflow post. Thanks



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