Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined by $$f(x) = \left\{\begin{matrix} \ln(e+x), & x\in\mathbb{N}\\ x^3, & x\in(-\infty, 0]\\ e^{-x}, & x\in(0,\infty)\setminus\mathbb{N} \end{matrix}\right.$$
Is function $f$ measurable on sigma algebra $m = \left \{ A \subseteq \mathbb{R} | A\textrm{ is countable set or } \mathbb{R}\setminus A \textrm{ is countable set} \right \}$?
What should I do in case like this one? Should I say: $\mathbb{N}$ is countable, neither $(-\infty, 0]$ nor $\mathbb{R}\setminus(-\infty,0]=(0, \infty)$ are countable, and also neither $(0, \infty)\setminus\mathbb{N}$ nor $\mathbb{R}\setminus(0,\infty)\setminus\mathbb{N}$ are countable. So, we can deduce that $f$ is not measurable on $m$. Is this the right way of thinking? Thank you.
$f^{-1}((-\infty,0))=(-\infty,0]$ which is not in the sigma algebra.