Lemma: Separated-support witness

There exist nonnegative $f,f^{\prime },g\in {C_c}^{\infty }(M_{+})$ with pairwise disjoint supports and

Their supports may also be chosen disjoint from ${\Sigma }_0$.

Proof. Choose $S,T,T^{\prime }>0$ pairwise distinct and ${\mathbf{n}}_0\in S^2$. Consider

These three spacetime points are distinct. Choose nonnegative smooth bumps $g,f,f^{\prime }$, respectively positive near these points, with sufficiently small pairwise disjoint supports contained in $M_{+}$.

For $y$ near $y_0$ and $\mathbf{n}$ near ${\mathbf{n}}_0$, the rays in (eq:ray) meet the regions where $f$ and $f^{\prime }$ are positive. Hence both ray transforms are positive on an open set on which $g$ is positive. Formula (eq:residue) then gives the result.