Corollary: Retained-event and TV margins

If the complete ideal and implemented outputs differ by trace distance at most $e$, then the probability of any retained event differs by at most $e$. In particular, an event of ideal probability zero has actual probability at most $e$.

If two ideal record laws have gap $\Delta$, and their implementations have complete output errors at most $e_0,e_1$, then

Proof. An event is an effect on the compared output. The variational characterization of trace distance bounds its probability difference. For the second claim, take the record marginals and apply the reverse triangle inequality.