  The functional renormalisation group equation is derived in a mathematically
rigorous fashion in a framework suitable for the Osterwalder-Schrader
formulation of quantum field theory. To this end, we devise a very general
regularisation scheme which retains reflection positivity as well as the
infinite degrees of freedom including smoothness. Furthermore, it is shown how
the classical limit is altered by the regularisation process leading to an
inevitable breaking of translation invariance. We also give precise conditions
for the convergence of the obtained theories upon removal of the
regularisation.
