# "trivial" measurability question

#### detalosi

Hey,

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.