  We address the question of distance record-setting by a random walker in the
presence of measurement error, $\delta$, and additive noise, $\gamma$ and show
that the mean number of (upper) records up to $n$ steps still grows universally
as $< R_n> \sim n^{1/2}$ for large $n$ for all jump distributions, including
L\'evy flights, and for all $\delta$ and $\gamma$. In contrast to the universal
growth exponent of 1/2, the pace of record setting, measured by the pre-factor
of $n^{1/2}$, depends on $\delta$ and $\gamma$. In the absence of noise
($\gamma=0$), the pre-factor $S(\delta)$ is evaluated explicitly for arbitrary
jump distributions and it decreases monotonically with increasing $\delta$
whereas, in case of perfect measurement $(\delta=0)$, the corresponding
pre-factor $T(\gamma)$ increases with $\gamma$. Our analytical results are
supported by extensive numerical simulations and qualitatively similar results
are found in two and three dimensions.
