I am considering a function \(\displaystyle u: \mathbb{N} \to \mathbb{R}\) on \(\displaystyle (\mathbb{N}, \mathcal{P}(\mathbb{N}))\).

By definition if for all \(\displaystyle A \in \mathcal{B}(\mathbb{R})\) we have that \(\displaystyle u^{-1}(A) \in \mathcal{P}(\mathbb{N}) \) then \(\displaystyle u\) is measurable.

Take an \(\displaystyle A \in \mathbb{R}\). Then by definition of preimage we have that \(\displaystyle u^{-1}(A) \in \mathbb{N}\). Since \(\displaystyle \mathbb{N} \subseteq \mathcal{P}(\mathbb{N}) \) then \(\displaystyle A \in \mathcal{P}(\mathbb{N})\).

My question is: what if i choose an \(\displaystyle A \in \mathcal{B}(\mathbb{R})\) where \(\displaystyle A \notin \mathbb{R}\). How do I guarantee that \(\displaystyle u^{-1}(A) \in \mathbb{N}\)?

Thanks.