How can I prove that these two statements relating to a measurable function are equivalent?

Question

Let $(\Omega, \Sigma)$, $(\mathbb{R}, \mathcal{B})$ be the two measurable space. Let $f:\Omega\rightarrow \mathbb{R}$ is a measurable function. Then the following statements are equivalent:

(1) $f^{-1}(A)=\{\omega\in\Omega: f(\omega)\in A\}\in \Sigma, \forall A\in \mathcal{B}$.

(2) $\{\omega\in\Omega: f(\omega)\in (-\infty, x]\}\in \Sigma,\forall x\in\mathbb{R}$.

I don't have a clue about how the above two statements are equivalent. Would anyone care to shed some light on it?

Answer 1

Hint

I suppose that $\mathcal B$ is the Borel $\sigma -$algebra. The implication $(1)\implies (2)$ is clear. For the converse, let $\mathcal A:=\{B\mid f^{-1}(B)\in \Sigma\}$. It's a $\sigma -$algebra that contains all sets of the form $(-\infty ,x]$. Therefore $$\mathcal B=\sigma \{(-\infty ,x]\mid x\in \mathbb R\}\subset \mathcal A.$$