In his 1981 paper "Quantum geometry of bosonic strings" Polyakov defines a measure for the summation over continuous surfaces. This measure must count all surfaces of a given area with the same weight. This means that if we have a transformation $\Omega$ which maps a surface $S$ to a surface $\widetilde{S}$ such that $A(S) = A(\widetilde{S})$ we must have for any functional $\phi[S],
$$\int d\mu(S) \phi(S) = \int d\mu(S) \phi(\widetilde{S}).$$
He claims that this leads to the following measure (for which he has no derivation):
$$\int d\mu(S) \phi(S) $$ $$=\int[Dg_{ab}(\xi)]\exp(-\lambda\int\sqrt{g}d^2\xi)$$ $$\times \int Dx(\xi)[\exp(-\frac{1}{2}\int_D\sqrt{g}g^{ab}\partial_a x_\mu\partial_b x_\mu d^2\xi)]$$ $$\times \phi[x(\xi)],$$ where $D$ is a unit disk in the $\xi$-plane.
My question: how to derive this measure from the general considerations at the start?
Possibly related: Integration measure for Polyakov's path integral
