Suppose that we have a Lebesgue measurable function $f: \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$, and consider some distribution, say the standard $n$-dimensional Gaussian $\mathcal{N}(0,1)$. Define the function $g:\mathbb{R}^n \rightarrow \mathbb{R}$ for every $p \in \mathbb{R}$ as $$g(p) := maj_{x \sim \mathcal{N}(0,1)} f(x,p)$$ as the most probable outcome of $f(x,p)$ when $x$ is sampled from $\mathcal{N}(0,1)$ and $p$ is fixed. That is $maj_{x \sim \mathcal{N}(0,1)}$ is the weighted majority according to the normal Gaussian. Also suppose that this majority is well-defined, that is, for every $p$ there exists some $x$ such that $\Pr_{x \sim \mathcal{N}(0,1)}[f(x,p)] \geq 1/2$
I am wondering if this function $g$ is measurable, and if not, under what conditions it can be made measurable. Although this seems like a pretty simple question, I haven't been able to find any information on it and have been unable to prove it myself after significant effort.
Thank you for the help.
My approach so far has been to define the measurable function $P_p: \mathbb{R}^n \rightarrow \mathbb{R}$ such that for every $A$, $\int_{A} P_p(x) \lambda(dx)$ is the probability that $f(p,x)$ takes value in $A$. Then it follows that $g(p) = argmax_x P_p(x)$. I am not sure if this is helpful.