Fix an infinite number $\omega$, and a finite number $n$. Let
$$
\Omega = \left\{\frac{k}{\omega}: k=1,2,\ldots,n\right\},
$$
then $\Omega$ is a subset of the infinitesimal in $[0,1]$.
Is it possible to construct the Loeb measure associated to
$$
X = (\Omega,\sigma(\Omega),\text{ counting measure}),
$$
where $\sigma(\Omega)=2^\Omega$ is the power set of $\Omega$.
That is
$$
\text{$X$ is a uniform random variable taking value in $\Omega$}.
$$
From this question, I should look for a first order description of $\sigma(\Omega)$.
But I don't see such description.