# Given $U\subseteq\mathbb{C}$ an open set and $f : X\rightarrow \mathbb{C}$ measurable then $f^{-1}(U)$ is measurable

Given the measurable space (X,$$B$$) where $$B$$ is a $$\sigma$$-algebra, I want to show that if $$f : X\rightarrow \mathbb{C}$$ is measurable (in the sense that, there exists a sequence of simple functions such that $$lim_{n\rightarrow\infty}f_{n}(x)=f(x)$$ for every $$x\in X$$) then $$f^{-1}(U)\in B$$.

I can get that for every single $$x\in f^{-1}(U)$$ there exists an $$N\in\mathbb{N}$$ such that $$f_{n}(x)\in U$$ for $$n\geq N$$

Also I can prove that as $$f_n$$ are simple functions then $$f_n^{-1}(U)\in B$$ but I am struggling trying to prove a relationship between $$f^{-1}(U)$$ and $$f_n^{-1}(U)$$.

My idea is trying to write:

$$f^{-1}(U) = \bigcap_{n\in\mathbb{N}}f_n^{-1}(U)$$

If that was true then I could write $$f^{-1}(U)$$ as the intersection of sets of $$B$$ and given that $$B$$ is a $$\sigma$$-algebra it would conclude the proof.

Any correction or idea will be welcomed.

• Welcome to Stack Exchange Mathematics. – Ramiro May 4 at 19:15
• I posted an answer to your question. Let me know if you have any questions regarding it. – Ramiro May 4 at 19:16

Since $$U$$ is open and $$\lim_{n\rightarrow\infty}f_{n}(x)=f(x)$$ for every $$x\in X$$, we have that, for all $$x$$:

$$f(x) \in U \iff \textrm{ there is } k\in\mathbb{N} \textrm { such that, for all } n\in\mathbb{N}, \; n>k,\; f_n(x)\in U$$

So in other words:

$$x\in f^{-1}(U) \iff \textrm{ there is } k\in\mathbb{N} \textrm { such that, } x\in \bigcap_{n>k}f_n^{-1}(U)$$

It follows that

$$f^{-1}(U) = \bigcup_{k\in\mathbb{N}} \bigcap_{n>k}f_n^{-1}(U)$$

Since $$f^{-1}(U)$$ as the union of intersections of sets of $$B$$ and $$B$$ is a $$\sigma$$-algebra, we have that $$f^{-1}(U)$$ is a set of $$B$$, which concludes the proof.

• Thank you so much Ramiro. I am still not very good with rewriting sets as unions and intersections as this is my first time studying meaure theory. This idea might help me in other exercises so thanks. – Antonio Gonzalez May 5 at 1:21