$L_{\infty}$ not separable: Find nested subsets of $\Omega \subset \mathbb{R}^d$ with positive finite Lebesgue measure?

Question

I'm trying to show that for any positive, finitely (Lebesgue) measured set $\Omega \subset \mathbb{R}^d$ the space of functions $L_{\infty}(\Omega)$ is not separable.

I have the argument down for $\mathcal{l}_{\infty}(\mathbb{N})$ using characteristic functions $\chi_J$ of arbitrary subsets $J \subset \mathbb{N}$. The analog in $L_{\infty}$ requests me to construct a nested sequence of subsets $\Omega_k \subset \Omega$ where each $\Omega_{k-1} \setminus \Omega_{k} $ is nonempty and has positive finite measure.

My intuition wants to find an interior point $x \in \Omega$ and use $\Omega_k = B_{1/k}(x) \cap \Omega$ and be done with it but $\Omega = \Big( \mathbb{R} \setminus \mathbb{Q} \Big) \cap [0,1]$ is a counterexample of the existence of such $x$.

Help/hints/solutions to my problem(s)?

Answer 1

Observe that the map $g: r \mapsto \lambda(B_r(0) \cap \Omega)$ is a continuous, nondecreasing map. Simply choose $r_i \in g^{-1}\left(\frac{\lambda(\Omega)}{i}\right)$ and $\Omega_i = B_{r_i}(0) \cap \Omega$.