Proposition: Coordinate convergence is not total-variation arrival convergence

If $t_j\to t$ with $t_j\ne t$, the deterministic arrival records ${\delta }_{t_j}$ and ${\delta }_t$ have total variation distance one. A fixed deadline can therefore have discontinuous acceptance.

Proof. The singleton $\{t\}$ separates the point masses. For $t_j=t+1/j$, the event $(-\infty ,t]$ has probabilities zero and one.