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Let $f:\mathbb{R}^n\to\mathbb{R}$ be an integrable function. Then for every $\epsilon>0$, there exists a compact set $K$ depending on $\epsilon$ such that $$ \int_{\mathbb{R}^n\setminus K}|f|<\epsilon. $$ I am curious to know if the above compact set $K$ can be chosen arbitrarily small.

Can someone please explain in detail? Thank you.

Mathguide
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  • Related with a little extra effort on your part you can get what you need from the link. – Mittens Apr 11 '22 at 19:01
  • Thanks, but still its bothering me, if its possible or not to choose small measure $K$. Note here the integration is outside $K$. – Mathguide Apr 11 '22 at 19:27
  • You compact set has to be "large" enough to contain most of the mass over which $f$ is integrable. That is why the statement is as it is. If your $K$ is small (in mass) you region of integration approaches the whole support of $f$. – Mittens Apr 11 '22 at 19:29
  • Thats what I also thought. So the compact set $K$ cannot be small in measure in the question, right? – Mathguide Apr 11 '22 at 19:31
  • If you want to make the integral of over the complement. no your $K$ cannot be made small (relative to the support of $f$). – Mittens Apr 11 '22 at 19:33

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