Proposition: A delayed nonzero experiment

For sufficiently small $\delta >0$, the source and readout supports are strictly time separated and

Thus the product experiment is prescription independent and nonzero for sufficiently small $\delta$.

Proof. The support of $f$ has time coordinate at least $t_0-a>0$. For $\delta <(t_0-a)/2$, the source occurs strictly earlier, and source and readout do not overlap. Therefore

Mollification converges to $f$ in every smooth-test seminorm on one common compact support. Distributional continuity and Theorem thm:response-invariant prove the claim.