Let us consider the real Hilbert space $H:=L^2\bigl(\mathbb{R}^n, \mathbb{R}^n\bigr)$ and "any" centered Gaussian measure $d\mu$ on it.

Next, denote a generic element of $H$ by the column vector $f(x)=\bigl[f_1(x), \cdots, f_n(x) \bigr]^T$.

Then, is it true that any $(ij)$-entry of the $2$-point correlation function for $\mu$, given by \begin{equation} C_{ij}(x,y):=\int_{H} f_i(x) f_j(y) d\mu(f) \end{equation} in facts depends on $x-y$ only?

That is, do we have \begin{equation} C_{ij}(x,y)=C_{ij}(x-y) \end{equation} for all $i,j=1,2,\cdots, n$?

I encountered such statements in the context of free scalar theories in QFT. However, I wonder if the property holds true in general.

Could anyone please help me?