Stein's Lemma and Metrics via Stein's Eq

Stein's Lemma Stein's Equation Let $\mathcal{H}$ be a seperating class and let $h \in \mathcal{H}$. Let $f_h$ be the solution to the Stein Equation associated with $h$. Then \[ f'_h(x)-xf_h(x) = h(x)-\mathbb{E}(h(N)) \] Let $F$ be a real valued random variable, then \[ f'_h(F)-Ff_h(F) = h(F)-\mathbb{E}(h(N)) \] Taking expectations on both sides \[ \mathbb{E}(h(F))-\mathbb{E}(h(N)) = \mathbb{E}(f'_h(F)-Ff_h(F)) \] Taking absolute values \[ | \mathbb{E}(h(F))-\mathbb{E}(h(N))| = |\mathbb{E}(f'_h(F)-Ff_h(F))| \] Taking a $\sup$ over $\mathcal{H}$, \[ d_{\mathcal{H}}(F,N)= \sup_{h \in \mathcal{H}} | \mathbb{E}(h(F))-\mathbb{E}(h(N))| = \sup_{h \in \mathcal{H}} |\mathbb{E}(f'_h(F)-Ff_h(F))| \] Metric on the Space of Probability Measures