0

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?

Community
  • 1
Borol
  • 491
  • dominated convergence theorem? – tired Jan 03 '17 at 13:38
  • 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. – Στέλιος Jan 03 '17 at 13:41

1 Answers1

2

$f_m$ 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.

Chappers
  • 67,606