If a real measurable function $f : E \rightarrow \mathbb{R}\ \cup \{\infty\}$ is finite almost everywhere (i.e. $f^{-1}(\{\infty\})$ has measure zero) and $E \subset \mathbb{R}$ is of finite Lebesgue measure, then $\forall \varepsilon > 0\ \exists\ F \subset E$ with $|E \setminus F| \leq \varepsilon$ and $f$ bounded in $F$.
I supposed that no such $F$ exists, and I tried to prove that $f = \infty$ in a set with positive measure, but without success. Any help? :)