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Is the pointwise maximum of two Riemann integrable functions Riemann integrable?

Let $f$ and $g$ be two integrable real functions. Is this leads that $\max\{f,g\}$ is integrable too?

Any proof?


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marked as duplicate by J. M., Srivatsan, Asaf Karagila, Jonas Meyer, t.b. Dec 1 '11 at 6:49

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

If you mean Riemann integrable, it was answered in this question:… – Martin Sleziak Nov 27 '11 at 13:08
I voted to close as duplicate, but I might be mistaken. The OP has not clarified whether Riemann or Lebesgue integration is intended. [Future voters: Please wait for some clarification from the OP.] – Srivatsan Nov 27 '11 at 15:16
@Martin, thanks, I had forgotten... – Did Nov 27 '11 at 22:40

$\max (f,g) = (f+g + |f-g|)/2$, so in the Lebesgue theory max(f,g) is integrable because linear combinations and absolute values of integrable functions are integrable.

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Better to say: Because of this identity, it suffices to prove the special case: If $f$ is integrable, then $|f|$ is integrable. – GEdgar Nov 27 '11 at 17:53

$$ |\max(f,g)|\leqslant\max(|f|,|g|)\leqslant|f|+|g| $$

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For full credit, needs more explanation. – GEdgar Nov 27 '11 at 17:52
GEdgar, thanks for your interest but I like it as is. Kind of similar to the point @Srivatsan made in a comment to the post, if you like. (What do you call full credit, by the way?) – Did Nov 27 '11 at 18:03

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