The density principle for orbital integrals

Let $S(F_0)$ be the symmetric space and let $\mathfrak{s}(F_0)$ be its Lie algebra. Consider the orbital-integral distributions $f \ifbool{@display}{\longmapsto}{\shortmapsto} \operatorname{Orb}(\gamma,f)$ for regular semi-simple $\gamma\in S(F_0)$, and their Lie-algebra analogues. Density principle. The orbital integrals $\operatorname{Orb}(\gamma,\cdot)$ for all regular semi-simple $\gamma$ span a weakly dense subspace of the space of $(H'(F_0),\eta)$-invariant distributions on $S(F_0)$. The same assertion holds for $\mathfrak{s}(F_0)$. This principle is used to pass between the two parts of the arithmetic-transfer conjectures; its status is not resolved in the supplied text.