I am studying measure theory and read this following sentence:

Suppose $\mu(E)<\infty$ and $f(x)$ is a measurable function on $E$. Then for any $0<p_0<\infty$, $\lim_{p\rightarrow p_0^-}\left\Vert f\right\Vert_p=\left\Vert f\right\Vert_{p_0}$

Does anyone knows why this is true?

  • 2
    $\begingroup$ Is $p_0$ really supposed to be greater than 0 or actually greater 1? If $p_0 > 1$ you can at least do some stuff with the Hölder inequality. If only $p > 0$, I'm not so sure about that. $\endgroup$ – Lukas Miristwhisky Nov 8 at 18:20
  • 1
    $\begingroup$ I would say that the overarching reason it is true is just that exponentiation is continuous. Of course that by itself isn't enough to prove it, but it is certainly where one should start looking. $\endgroup$ – Paul Sinclair Nov 9 at 3:47

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