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Proposition: Let $f_1, ..., f_n$ real valued integrable lebesgue functions in $(X, \mathfrak{M}, \mu)$ meausure space. Then $max\{f_1, ..., f_n\}$ is measurable.

proof: We have that $max\{f_1, ..., f_n\} = max\{f_1, max\{f_2, max\{...max\{f_{n-1}, f_n\}...\}\}\}$; as max of integrable lebesgue functions is integrable lebesgue function then $max\{f_1, ..., f_n\}$ is integrable lebesgue function.

There is some mistake?

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    No mistake, but then this seems so trivial unless you explain why the maximum of two functions $\max \{ f,g\}$ is integrable whenever $f,g$ are. I know the answer, but want to make sure that you know it too. – 2017-01-29
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    $max\{f, g\} = \dfrac{f + g + |f - g|}{2}$ – 2017-01-29
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    Perfect. That's why. All right, your question is done! – 2017-01-29

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