  The exact renormalization group equation for pure quantum gravity is derived
for an arbitrary gauge parameter in the space-time dimension $d=4$. This
equation is given by a non-linear functional differential equation for the
effective average action. An action functional of the effective average action
is approximated by the same functional space of the Einstein-Hilbert action.
  From this approximation, $\beta$-functions for the dimensionless Newton
constant and cosmological constant are derived non-perturbatively. These are
used for an analysis of the phase structure and the ultraviolet non-Gaussian
fixed point of the dimensionless Newton constant. This fixed point strongly
depends on the gauge parameter and the cutoff function. However, this fixed
point exists without these ambiguities, except for some gauges. Hence, it is
possible that pure quantum gravity in $d=4$ is an asymptotically safe theory
and non-perturbatively renormalizable.
