In section 3.4 of Blumenhagen's Basic Concepts in String Theory, where path integral quatization is presented, and we are given the partition function for the Polyakov's path integral $$Z=\int \mathscr{D} h \mathscr{D} X e^{i S_P[h, X]}\tag{3.63}$$
(I think normally this would be Wick rotated to Euclidian spacetime, which effectively replace the $i$ with a minus sign.)
The integration measure is defined as
$$\begin{aligned} & \|\delta h\|^2=\int d^2 \sigma \sqrt{-h} h^{\alpha \beta} h^{\gamma \delta} \delta h_{\alpha \gamma} \delta h_{\beta \delta}, \\ & \|\delta X\|^2=\int d^2 \sigma \sqrt{-h} \delta X^\mu \delta X_\mu\end{aligned}\tag{3.64}$$ the same as in Polyakov's original paper. But the author states this integration measure is derivable from the following Gaussian integral $$1=\int \mathscr{D}(\delta X) e^{-\frac{1}{4 \pi \alpha^{\prime}}\|\delta X\|^2}\tag{3.65}$$ and other variables are defined "likewise".
I am not sure how $(3.64)$ follows from this Gaussian integral definision and I did not find this in Polyakov's paper. Although it is apparent if we make the replacement $||\delta X||^2\rightarrow h^{\alpha\beta} \partial X^\mu_\alpha X^\nu_\beta \eta_{\mu\nu}$ we would recover the Polyakov action on the exponent. Also, how would one define the measure for the metric implicitly through a Gaussian integral similar to $(3.65)$?
