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Possible Duplicate:
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?

Thanks

  • 0
    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.]2011-11-27

2 Answers 2

16

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

  • 1
    Better to say: Because of this identity, it suffices to prove the special case: If $f$ is integrable, then $|f|$ is integrable.2011-11-27
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$ |\max(f,g)|\leqslant\max(|f|,|g|)\leqslant|f|+|g| $

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    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?)2011-11-27