Interesting Measure and Integration Question

Let (\Omega,\mathcal{A},\mu) be a measure space. Let f\in L^p and \epsilon>0. Prove that there exists a set E\in\mathcal{A} with \mu(E)<\infty, such that \int_{E^c} |f|^p<\epsilon.

Solution:

The solution strategy is to use simple functions (common tactic for measure theory questions).

Let 0\leq\phi\leq |f|^p be a simple function such that \int_\Omega (|f|^p-\phi)\ d\mu<\epsilon.

Consider the set E=\{\phi>0\}. Note that \int_\Omega \phi\ d\mu\leq\int_\Omega |f|^p\ d\mu<\infty. Hence each nonzero value of \phi can only be on a set of finite measure. Since \phi has only finitely many values, \mu(E)<\infty.

Then,

\begin{aligned} \int_{E^c}|f|^p\ d\mu&=\int_{E^c} (|f|^p-\phi)\ d\mu +\int_{E^c}\phi\ d\mu\\    &\leq \int_\Omega (|f|^p-\phi)\ d\mu+0\\    &<\epsilon    \end{aligned}

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