  We consider the voter model on Z, starting with all 1's to the left of the
origin and all 0's to the right of the origin. It is known that if the
associated random walk kernel p has zero mean and a finite r-th moment for any
r>3, then the evolution of the boundaries of the interface region between 1's
and 0's converge in distribution to a standard Brownian motion (B_t)_{t>0}
under diffusive scaling of space and time. This convergence fails when p has an
infinite r-th moment for any r<3, due to the loss of tightness caused by a few
isolated 1's appearing deep within the regions of all 0's (and vice versa) at
exceptional times. In this note, we show that as long as p has a finite second
moment, the measure-valued process induced by the rescaled voter model
configuration is tight, and converges weakly to the measure-valued process
1_{x<B_t}dx, t>0.
