This relates to the Bochner's theorem stated in the link:
https://en.wikipedia.org/wiki/Bochner%27s_theorem
My question is related to the unique probability measure μ on G. I want to express the term $dμ=g1(x)dx$ (In terms of another function say $g1$) My question what kind of $g1$ is applicable? Any comments would be appreciated.
Here is the finite dimensional version of Bochner's Theorem. Maybe this will help you. If $f = (f_n)_{0 \le n \le N-1}$ is a positove definite sequence, then there exists another sequence $g = (g_n)_{0 \le n \le N-1}$ such that $f$ is the discrete Fourier transform of $g$, and $g_n > 0$. Positive definite means $$ \sum_{i,j=0}^{N-1} f_{(i-j)\bmod N} x_i x_j > 0 $$ whenever $x = (x_n)_{0 \le n \le N-1}$ isn't identically zero. Equivalently the circulant matrix https://en.wikipedia.org/wiki/Circulant_matrix built from $f$ is a positive definite matrix.
In the finite dimensional case, it is easy to construct $g$ - just take the inverse discrete Fourier transform. But in the general situation, you don't know a priori that the inverse Fourier transform exists.