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I want to compute that $\lim\limits_{n\to\infty}\int_{[0,1]}\frac{nx^{n-1}}{1+x} \, d\lambda(x)=\frac{1}{2}$, where $\lambda$ is the Lebesgue measure.

I want to use Lebesgue dominated convergence theorem, but I can't find a uniform bound for $|\frac{nx^{n-1}}{1+x}|$, and the limit function tends to be $\begin{cases}0 & x<1 \\\infty & x=1\end{cases}.$ How do I deal with this integral?

Marios Gretsas
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2 Answers2

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Make the substitution of variable $u = x^n$ which implies $du = n x^{n-1}dx$ to get $$\int_0^1 \frac{nx^{n-1}}{1+x} \ dx = \int_0^1 \frac{1}{1+ u^{1/n}} \ du$$

And now, you can apply Lebesgue dominated convergence theorem as $$0 \le \frac{1}{1+ u^{1/n}} \le 1$$ and $\lim\limits_{n \to \infty} \frac{1}{1+ u^{1/n}} = 1/2$ for $u \in (0,1]$.

mathcounterexamples.net
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  • We didn't officially treat substitutation of variables yet in a measure theoretic version –  Oct 22 '19 at 13:59
  • OK. How did you define the integral of a function in your course then? – mathcounterexamples.net Oct 22 '19 at 14:01
  • As the difference between the positive and negative part of the function, then the integral of the positive part is the supremum over the integrals of all step functions which lie under the positive part. –  Oct 22 '19 at 14:02
  • Won't be easy to use LDC theorem if you use only that. You're not even suppose to know fundamental theorem of calculus? – mathcounterexamples.net Oct 22 '19 at 14:11
  • We do know about differentiation under the integral sign: If $u:(a,b)\times X\to\mathbb{R}$ is integrable and differentiable with respect to $t$ and $|\partial_tu(t,x)|\leq|w(x)|$ for all $(t,x)$ with $w$ integrable, then I can change differentiation and integration order. –  Oct 22 '19 at 14:14
  • The integrand looks like some derivative, I must admit –  Oct 22 '19 at 14:15
  • @James ...all the functions in the sequence are Riemman integrable on $[0,1]$ because of continuity and their Riemman integral coincides with the Lebesgue integral..so you can do change of variables treating the integrals as Riemman integrals..so there is not any case of measure theoretic change of variables. – Marios Gretsas Oct 22 '19 at 14:31
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Dominated convergence allows to interchange limit and integral.The integral of the pointwise limit is zero. However, the limit of the integrals is non-zero. So dominated convergence cannot work for this example.

daw
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