Let $ f \in L_p(X, \mathcal{A}, \mu)$, $1 \leq p < \infty$, and let $\epsilon > 0$. Show that there exists a set $E_\epsilon \in \mathcal{A}$ with $\mu(E_\epsilon) < \infty$ such that if $F \in \mathcal{A}$ and $F \cap E_\epsilon = \emptyset$, then $\|f \chi_F\|_p < \epsilon$.
I was wondering if I could do the following:
$ \int_X |f|^p = \int_{X-E_{\epsilon}}|f|^p + \int_{E_\epsilon}|f|^p \geq \int_{F}|f|^p + \int_{E_\epsilon}|f|^p $
$ \int_{X}|f\chi_F|^p=\int_{F}|f|^p \leq \int_X |f|^p- \int_{E_\epsilon}|f|^p < \epsilon$
Please just point out any fallacies within my logic as I have found this chapter's exercise to be quite difficult, so any help is appreciated.