Lemma: Let $ (X,\mathcal{A},\mu) $ be a finite measure space, and let $ f $ be a measurable function on E. Then

$ \lim_{n \rightarrow \infty} \int_X |f|^n = \mu\left\{|f|=1\right\} + \infty * \mu\left\{|f|>1\right\} $

where we interpret $ \infty * 0 = 0 $.

Proof: $ \int_X |f|^n = \int\limits_{\left\{|f|>1\right\}}\!\!\!\!\!|f|^n + \!\! \int\limits_{\left\{|f|=1\right\}}\!\!\!\!\!|f|^n + \!\! \int\limits_{\left\{|f|<1\right\}}\!\!\!\!\!|f|^n $

We have

$ \int\limits_{\left\{|f|=1\right\}} \!\!\!\!\! |f|^n \rightarrow \mu\left\{|f|=1\right\} $,

$ \int\limits_{\left\{|f|>1\right\}} \!\!\!\!\!|f|^n \rightarrow \infty * \mu\left\{|f|>1\right\} $ by Fatou (or Monotone Convergence Theorem), and

$ \int\limits_{\left\{|f|<1\right\}} \!\!\!\!\! |f|^n \rightarrow 0 $ by the Bounded Convergence Theorem, since $ \mu (X) < \infty $. $ \square $

Remark: The hypothesis that $ \mu(X) < \infty $ cannot be omitted, as $ f(x) = \frac{x}{x+1}, X = [0,\infty) $ shows. However, if we require that $ |f|^k \in L^1 $ for some k, then the third equality follows from the Monotone Convergence Theorem, and the Lemma holds under this weaker hypothesis.


a) $ \lim_{n \rightarrow \infty} \int_0^\pi \sin^n(x)dx = 0 $

b) $ \lim_{n \rightarrow \infty} \int_0^\pi 2^n\sin^n(x)dx = \infty $

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