Measurability of a function on sub sigma algebra

Let $Omega$ be a compact Hausdorff space in $mathbb{C}^2$. Let $sigma_Omega$ be the Borel sigma algebra on $Omega$. Let $f: Omegalongrightarrowpartial mathbb{D}$ be a non constant continuous function. Let $sigma_{partial mathbb{D}}$ be the Borel sigma algebra on $partial mathbb{D}$(Unit circle on the complex plane). Now consider the sigma algebra $sigma_0={f^{-1}(A): ;Ain sigma_{partial mathbb{D}}}subset sigma_Omega$.

Now consider the function $g: Omegalongrightarrowmathbb{C}$ as $g(z_1,z_2)=z_1^mz_2^n(bar{z_1})^p(bar{z_2})^q$, where $m,n,p,qinmathbb{N}.$

Now as $g$ is continuous, it is clearly measurable w.r.t $sigma_Omega$ but. will it be measurable w.r.t $sigma_0$?

Is there any additional condition that can be put on $f$ , so that the above holds?