Generally speaking, the Double Jeopardy Clause precludes the government from appealing an acquittal (whether an acquittal by a jury or by a judge). Once you're acquitted, you're off the hook, even if the trial judge erred in instructing the jury, in deciding whether there was sufficient evidence, in admitting evidence, in refusing to admit evidence, or the like.
But I just learned that, for the last 100 years, Colorado has allowed the prosecution to appeal acquittals, even though the appeal couldn't affect the result. Indeed, the bottom line decision wouldn't be "judgment affirmed" or "judgment reversed" (since the judgment of acquittal couldn't be reversed)—rather, it would be "judgment approved" or "judgment disapproved."
The prosecution's goal in such a case would presumably be to try to set a precedent that would foreclose similar errors in the future. Of course, if the judgment is approved because the appellate court agrees with the trial court, then this would set a precedent that would encourage such trial court decisions in the future.
