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Possible Duplicate:
Limit of $L^p$ norm

I was asked to show:

Assume $|f|_{r}<\infty$ for some $r<\infty$. Prove that $$ |f|_{p}\rightarrow |f|_{\infty} $$ as $p\rightarrow \infty$.

I am stuck in the situation that $|f|_{p}<\infty$ for all $r, but $|f|_{\infty}=\infty$ nevertheless. Could this happen? Imagine $f^{-1}(n,n+1)$ has measure $\frac{1}{n^{n}}$, for example. Then $|f|_{p}$ exists for any $p$, but $|f|_{\infty}=\infty$ nevertheless. However, I do not know how to show $f_{p}$ must be monotonely increasing in this case.

Could $f_{p}$ be fluctuating while $|f|_{\infty}=\infty$? I have proved that for $r, $|f|_{p}<\max (|f|_{r},|f|_{s})$. But this does not help to show $|f|_{p}$ does not fluctuate.

  • 1
    Could you write down the definition of $\vert f \vert_{\infty}$ in your post?2012-12-24
  • 0
    The usual definition that $f^{-1}(c,\infty)$ has measure 0, and $c$ is the inf of all such values possible.2012-12-24
  • 0
    I am sure this is a duplicate question, although cannot find it now.2012-12-24
  • 0
    This is from Rudin, so will not be surprising if it is a duplicate.2012-12-24
  • 2
    See [this](http://math.stackexchange.com/questions/242779/limit-of-lp-norm/242792) post.2012-12-24
  • 0
    Thanks! As I know I would not be the only one...:(2012-12-24

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