Measurable functions and Borel sets

Question

Show that function $f$ from $E$ to $R^*$ defined on a measurable set $E$ is measurable if and only if:

1. The sets $f^{-1}(\infty)$ and $f^{-1}(-\infty)$ are measurable and
2. For each Borel set $B$, $f^{-1}(B)$ is measurable.

Answer 1

Hint: One way is very easy. The two conditions imply measurability.

Suppose $f$ is measurable. Then $f^{-1} (\infty)=\cap_n [E \cap (f>n)]$ and $f^{-1} (-\infty)=\cap_n [E \cap (f>-n)^{c}]$.