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Let $p \in (0, \infty)$ and the sequence of functions $f_m : [0, 1] \rightarrow \Bbb R, m \in \Bbb N$,

$f_m(x) := ((1 - x^p)^{1 \over p})^m$.

Calculate

$\lim_{m \rightarrow \infty} \int_{[0, 1]} f_m(x) d\lambda(x)$.

How would I start approaching this problem? I remember that there is a theorem that allows me to change the limes and the integral under certain conditions, but it's been a while since I used it. Should I start with this?

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    dominated convergence theorem? – 2017-01-03
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    The theorem you are talking about is propably the Dominated Convergence Theorem of Lebesgue. Search it on wiki. Here the functions are continuous on the closed interval $[0,1]$ and therefore bounded. Moreover the measure of the interval (length in the case of intervals) is finite. Thus the functions are dominated. So you can apply the theorem. – 2017-01-03

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$f_m$ is dominated by the integrable function $1$. Since $(1-x^p)^{1/p}<1$ for almost every $x$ in the interval, raising it to the $m$th power makes it closer and closer to $0$ as $m$ increases. Then the Dominated Convergence Theorem gives $\int_{[0,1]} 0 \, d\lambda = 0$ as the answer.

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    Thank you very much! In order to apply the theorem, I would also have to show that the sequence of the functions is indeed integrable, right? – 2017-01-03
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    Yes. It is enough to show it's continuous in this case. – 2017-01-03
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    Isn't it just continuous as a composition of continuous functions? – 2017-01-03
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    Or: Isn't it integrable because it is bounded on $[0, 1]$? – 2017-01-03
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    Boundedness isn't good enough (Vitali sets show this), but yes, it's a composition of continuous functions, so it is cts. – 2017-01-03
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    Thank you very much for your effort! – 2017-01-03