In regression discontinuity (RD) with a running variable S crossing a known cutoff c , an unexpectedly small break magnitude is due to S being a mis-measured version of the genuine running variable G . Has all been lost, and is RD useless when G≠S ? This paper proves three things. First, when P ( G=S )=0, nonparametric RD identification fails. Second, when P ( G=S )>0, although the usual RD effect on the margin E (·| G=c ) is not nonparametrically identified, the “effect on the truthful margin” E (·| G=S=c ) is. Third, under a no-selection-problem assumption, the effect on the truthful margin becomes the effect on the margin; the no-selection-problem assumption is unnecessary, as long as the effect on the truthful margin is taken as a parameter of interest.