# $f \in L^p(\Omega) \Rightarrow$ that exist a ball $B_R \subset \mathbb{R}^n$ that $\int_{\Omega \backslash B_R}|f|^p < \varepsilon$.

Let $f \in L^p(\Omega)$ and $\varepsilon >0$. Show that exist a ball $B_R \subset \mathbb{R}^n$ such that $\int_{\Omega \backslash B_R}|f|^p < \varepsilon$.

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What is $\Omega$ ? An open set ? The ball has to be included in $\Omega$ ? Did you try with dominated convergence theorem ? – Ahriman Aug 29 '12 at 20:21
Also, you should ask a question rather than state an exercise. – Matt Aug 29 '12 at 20:34

Hint: $$\int_\Omega = \sum_{n=1}^\infty \int_{\Omega \cap \{x: n-1 \le |x| < n\}}$$