  We extend the epsilon multiplicity of ideals defined by Ulrich and Validashti
to epsilon multiplicity of filtrations, and show that under mild assumptions
this multiplicity exists as a limit. We show that in rather general rings, the
epsilon multiplicity of a Q-divisorial filtration is positive if and only if
the analytic spread of the filtration is maximal (equal to the dimension of the
ring). The condition that filtrations $\mathcal J\subset \mathcal I$ have the
same epsilon multiplicity is considered, and we find conditions ensuring that
the filtrations have the same integral closure.
