Lemma: Positive reflected observation measure

The map $\Psi$ in (eq:residue) is an embedding proper over compact subsets of $X$. The distribution $r$ is a positive locally finite measure of order zero, with support exactly $\mathcal{R}$. For product tests,

where

uses zero extension of $f$ outside $V$.

Proof. The observation times recover $t,s,t^{\prime }$. On the image,

Since $t+s>0$, these are smooth inverse formulae. They also prove immersion and injectivity.

On the inverse image of a compact subset of $X$, all times are bounded and separated from zero, $\mathbf{z}$ is bounded, and $\mathbf{n}$ lies in the compact sphere. The inverse image is closed in the resulting compact parameter region, proving properness there.

The parameter density is positive, so its pushforward is a positive locally finite measure. Every neighbourhood of an image point has inverse image of positive parameter measure; hence its support is the whole image. Changing $\mathbf{z}$ to $\mathbf{y}=\mathbf{z}-s\mathbf{n}$ in the defining integral gives (eq:productresidue).