Examples of nontrivial non-discriminatory functions
Let $n=1$. Let $\mu$ be the usual Lebesgue length measure on $[1/2,1]$, and let $\mu$ be the negative of the usual Lebesgue length measure on $[0,1/2]$.

In particular, Lebesgue measure is $|\mu|$.

Let $\mathcal U\subseteq [0,1]$ be a set of Lebesgue measure 0. (For instance, $\mathcal U$ could be the set of all numbers whose binary expansion does not satisfy the law of large numbers...)

Let


$\sigma(x)=1$ for $x\in\mathcal U$, and
$\sigma(x)=0$ for $x\not\in\mathcal U$.


Then given $y$ and $\theta$, there are two cases.

Case 1: $y\ne 0$. Then
$$|\mu|\{X:y^\top X+\theta\in\mathcal U\}=0,$$ hence
$$|\mu|\{X:\sigma(y^\top X+\theta)\ne 0\}=0,$$
so $\int \sigma(y^\top X+\theta)d|\mu|(x)=0$,
so $\int \sigma(y^\top X+\theta)d\mu(x)=0$.

Case 2: $y=0$. Then $\sigma(y^\top X+\theta)$ is the constant $\sigma(\theta)$, which satisfies
$$\int \sigma(\theta)\,d\mu(x)=\sigma(\theta)\int d\mu(x)=\sigma(\theta)\mu([0,1])=0.$$

So, $\sigma$ is non-discriminatory.