Hellinger bound for differences of expectations

If $f$ is a measurable function into a separable Banach space and has finite second moments under probability measures $\mu ,{\mu }^{\prime }$, then their Bochner integral expected values satisfyHere the Hellinger distance includes the factor $1/2$ in its square. For a proof, use $\nu =(\mu +{\mu }^{\prime })/2$, $p=d\mu /d\nu$, and $q=d{\mu }^{\prime }/d\nu$. Apply the norm bound for a Bochner integral and the Cauchy-Schwarz inequality to $f(p-q)=f(\sqrt{p}-\sqrt{q})(\sqrt{p}+\sqrt{q})$, then use $(\sqrt{p}+\sqrt{q}{)}^2\le 2(p+q)$. This explains why stability of a Bayesian posterior in Hellinger distance controls posterior expected values when the relevant second moments remain bounded.