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This is exercise 2 from chapter 15 in Bauer's book Measure and Integration Theory:

For an arbitrary measure space $(\Omega,\mathcal{F},\mu)$ and $1\leq p<\infty$ show that a real function $f$ on $\Omega$ is $p$ integrable if and only if $f\lvert f \rvert^{p-1}$ is integrable. (In the "if" direction measurability of $f$ itself is not part of the hypothesis.)

Since $\lvert f \rvert^{p}= \bigg\lvert f\lvert f \rvert^{p-1} \bigg\rvert $ the assertion clearly holds if $f$ is measurable. But in "if" direction how I show that measurability of $f$ follows from the measurability of $f\lvert f \rvert^{p-1}$?

Any help is greatly appreciated.

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1 Answer 1

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Hints: $|f|f|^{p-1}|=|f|^{p}$ is measurable. Composing with the continuous function $x \to x^{1/p}$ we see that $|f|$ is measurable. In particular $\{x: f(x)=0\}=\{x: |f(x)|=0\}$ is measurable. On the set $\{x: f(x)\neq 0\}$ we can write $f=\frac 1{|f|^{p-1}} f|f|^{p-1}$. Can you finish?

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  • $\begingroup$ Right! If I understand correctly the restriction $f'$ of $f$ to $\Omega'=\{ x : f(x) \neq 0 \} $ is $\Omega' \cap \mathscr{B}$ measurable. Thus the preimage of any borel set under $f$ must be either $\Omega' \cap B$ or $(\Omega' \cap B) \cup (\Omega \setminus \Omega' ) $ for some $B \in \mathscr{B}$, in either case an element of $\mathcal{F}$. $\endgroup$
    – Alphie
    May 31, 2020 at 9:05
  • $\begingroup$ Do you know Bauer's book? I plan on reading his book on probability theory as well. Do you think they are good books to learn the subject? $\endgroup$
    – Alphie
    May 31, 2020 at 9:21
  • $\begingroup$ Yes, Bauer is an excellent author. @Alphie $\endgroup$ May 31, 2020 at 9:23

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