Show that if a function $f:E \rightarrow \mathbb{R} $ satisfies $\{x \in E : f(x) \leq y \} \in \epsilon$ $\forall y \in \mathbb{R}$ is measurable from $(E,\epsilon)$ to $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$
Thoughts: I'm trying to think of a way to show that the inverse image of $f^{-1}(A)$ is in $\epsilon$ for all A in the Borrel sigma algebra, I know I could also show that $f^{-1}(A^{c})$ is in $\epsilon$ but I'm not really sure how to go about it for a function when all I know is it satisfies $\{x \in E : f(x) \leq y \} \in \epsilon$ $\forall y \in \mathbb{R}$