I am currently trying to gain a better understanding of the gauge fixing procedure used in chapter 5 of David Tong's notes.

Since the central charge of the Polyakov action for, say, the bosonic string is not zero the measure $\mathcal{D}X$ has non-trivial dependence on $g$ in $$Z[g]=\int\mathcal{D}X\exp(-S_{Pol}[X,g])$$ where $g$ is the metric on the string (worldsheet). Importantly it changes non-trivially under supposed gauge transformations of $g$. I will from now on denote it as $\mathcal{D}_gX$.

Now, when gauge fixing using the Faddeev-Popov method to calculate

$$Z=\frac{1}{\mathrm{Vol}} \int\mathcal{D}g\int\mathcal{D}_gX\exp(-S_{Pol}[X,g]).\tag{p.109}$$ Tong inserts $$1=\Delta_{FP}[g]\int\mathcal{D}\zeta\delta(g-g_0^\zeta)\tag{5.1}$$ where $\int\mathcal{D}\zeta$ is an integral over the gauge group and $g_0^\zeta$ is a 'reference' metric acted on by the gauge group element $\zeta$. One gets, by integrating in $g$ using the delta function, $$Z=\frac{1}{\mathrm{Vol}}\int\mathcal{D}\zeta\int\mathcal{D}_{g_0^\zeta}X\Delta_{FP}[g_0^\zeta]\exp(-S_{Pol}[X,g_0^\zeta]) .\tag{p.111}$$ At this point Tong changes all $g_0^\zeta$ to $g_0$ and uses $\frac{1}{\mathrm{Vol}}\int\mathcal{D}\zeta=1$.

While the Faddeev-Popov determinant and the Polyakov action are really gauge invariant, the integration measure clearly isn't. Why is this not mentioned or acknowledged in the text? Am I having a misunderstanding of the nature of the conformal anomaly?