  We derive a simple formula for the fluctuations of the time average around
the thermal mean for overdamped Brownian motion in a binding potential U(x).
Using a backward Fokker-Planck equation, introduced by Szabo, et al. in the
context of reaction kinetics, we show that for ergodic processes these finite
measurement time fluctuations are determined by the Boltzmann measure. For the
widely applicable logarithmic potential, ergodicity is broken. We quantify the
large non-ergodic fluctuations and show how they are related to a super-aging
correlation function.
