Borel measurable function $f : X \rightarrow \mathbb{C}$.

Question

In some old books I found the following definition:

"Given $f : X \rightarrow \mathbb{C}$, $f$ is a Borel measurable function if $f^{-1}(B)$ is a Borel set [of X] for any open set $B \subseteq \mathbb{C}$."

My question is: if we consider the Borel $\sigma-$algebra for $X$ and also for $\mathbb{C}$, is enough to check the condition of the definition to show that a function is measurable with respect to these two $\sigma-$algebras?

I would be truly grateful if you could give some references.

Thank you

Answer 1

Yes. Let $X$, $Y$ non-empty sets, $A$ a $ \sigma-$ algebra on $X$, $B$ a $ \sigma-$ algebra on $Y$ and $f:X \to Y$ a mapping. If $C$ is a collection of subsets of $B$ such that $\sigma(C)=B$, then we have:

$f$ is $(A,B)-$ measurable $ \iff f^{-1}(M) \in A$ for all $M \in C$.