IANAL, but in the capital case, yes for several reasons:

1. he may not have been sentenced to death in the 2nd case;
2. the jury may not have sentenced the real murderer to death in the 1st case since there are obviously different circumstances involved;
3. presumably there was evidence he committed the 1st murder as well.  Just because there is evidence he committed the 2nd murder doesn’t mean that A) he actually did it, or B) a jury would even convict him;
4. since our man is already dead, there would never be a trial for the 2nd murder, and never an opportunity for his defense to be mounted;
5. a trial has cathartic value for community.  In your example, this is thwarted for both murders;
6. what about the real perpetrator of the 1st crime? I suspect that a jury would be more likely to avoid sentencing him to death on the, “yeah we’re pretty sure we got the right guy this time, but let’s go a little easy here just-in-case” principle. 

Also, the crime against another person strikes me as fundamentally different from a misdemeanor against the state.  I think it’s OK to have a social policy regarding minor infractions that essentially says, “no harm? No foul.”  These minor-infraction laws aren’t always there to ensure the prohibited behavior is never exemplified; they’re there to provide a framework to resolve disputes in the case when a citizen’s behavior becomes a cause for dispute by either another citizen or the local constabulary.  

Anyway, if it’s a lying police officer, then an injustice also exists as the officer in question is now exposed as duplicitous to the fellow with the undeserved ticket.  He’ll make sure everyone he knows is aware, which will contribute to the eventual guilt-by-association that people love to exercise.  Look at the LAPD; a few rotten Mark Fuhrman apples and the entire force has a rotten-barrel image problem for a generation or more.  Next thing you know it’s riots in the streets and Terry gets that revolution he’s been hankering after.

So the moral of the story: forget arguing about the injustice of executing the wrong fellow and keep your eye on the traffic cops before they ruin the country.

squashed:

A man is convicted of murder, sentenced to death, and is hanged. After his death, new evidence emerges conclusively proving he had not committed the murder for which he was executed. However, evidence emerges in another case showing the executed man had murdered somebody else. Was his hanging unjust? (Pretent for a moment that capital punishment isn’t an all around bad idea.)

Does it change if we lower the stakes? A man was given a hefty speeding ticket for allegedly going 55 in a 35. Although he was actually following the speed limit, a judge believed the (lying) police officer an upheld the ticket. While he wasn’t speeding that time, he had driven 55 down the same road a few days previously but had not been caught. Was this an injustice?