If $f$ and $g$ integrable then $\max\{f,g\}$ is? [duplicate]

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

Any proof?

Thanks

-

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

If you mean Riemann integrable, it was answered in this question: math.stackexchange.com/questions/72844/… – 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.
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
$$|\max(f,g)|\leqslant\max(|f|,|g|)\leqslant|f|+|g|$$