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Question. Let $(X, M, m)$ be an arbitrary measure space. Let $f \in L^1_m(X)$. Then $$\lim_{n \rightarrow \infty} \int_X n \log \left( 1 + \frac{|f(x)|^2}{n} \right) \,dm(x) = 0.$$


Solution.

Observe that $$n \log \left( 1 + \frac{|f(x)|^2}{n} \right) \leq n \log \left( 1 + \frac{|f(x)|}{n} \right)^2 = 2n \log \left( 1 + \frac{|f(x)|}{n} \right) \leq 2n \frac{|f(x)|}{n} = 2|f(x)|.$$

Then $n \log \left( 1 + \frac{|f(x)|^2}{n} \right)$ is dominated by $2|f(x)|$.

Moreover, we have

$$\lim_{n \rightarrow \infty} \frac{\log \left( 1 + \frac{|f(x)|^2}{n} \right)}{1/n} = \frac{0}{0}.$$

So we apply L'Hopital's Rule and we obtain:

$$\lim_{n \rightarrow \infty} \frac{\log \left( 1 + \frac{|f(x)|^2}{n} \right)}{1/n} = \lim_{n \rightarrow \infty} \frac{\frac{-|f(x)|^2}{n^2} (-n^2)}{ 1 + \frac{|f(x)|^2}{n} } = |f(x)|^2.$$

Then by Lebesgue Dominated Convergence Theorem, we conclude that

$$\lim_{n \rightarrow \infty} \int_X n \log \left( 1 + \frac{|f(x)|^2}{n} \right) \,dm(x) = \int_X |f(x)|^2 \,dm(x).$$


Where is the proof wrong? Thanks.


Added Later. I found the mistake: $n \log \left( 1 + \frac{|f(x)|^2}{n} \right) \leq n \log \left( 1 + \frac{|f(x)|}{n} \right)^2$ is not true for all $n$. As @did suggested, dominating function should be $|f|^2$.

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    $\begingroup$ Your title does not seem to be correct, since the limit is not $0$ but the integral of $|f|^2$. $\endgroup$ – Julien Jan 6 '13 at 20:36
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    $\begingroup$ And please replace every $dm$ by $dm(x)$. $\endgroup$ – Did Jan 6 '13 at 23:02
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Let $g_n=n\log(1+|f|^2/n)$, then $\log(1+t)\leqslant t$ for every $t$, hence $g_n\leqslant|f|^2$. If $f$ is in $L^2$, this shows that $(g_n)$ is dominated. Since $g_n\to|f|^2$ pointwise, one gets $\int\limits_Xg_n\to\int\limits_X|f|^2$.

If $f$ is not in $L^2$, let $A(C)=\{|f|\leqslant C\}$ for some $C\gt0$, then $g_n\geqslant n\log(1+|f|^2/n)\mathbf 1_{A(C)}$. Furthermore, $\log(1+t)\geqslant t\log2$ for every $t\leqslant1$ hence, for every $n\geqslant C^2$, $g_n\geqslant(\log2)|f|^2\mathbf 1_{A(C)}$ and $\liminf\limits_{n\to\infty}\int\limits_Xg_n\geqslant\int\limits_{A(C)}(\log2)|f|^2$. This holds for every $C$ and $\int\limits_{A(C)}|f|^2$ can be made as large as one wants hence $\int\limits_Xg_n\to+\infty$.

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  • $\begingroup$ Your $n\log(1+|f|^2/n)\leqslant2|f|$ is wrong. $\endgroup$ – Did Jan 6 '13 at 19:38
  • $\begingroup$ Then the question is not correct? I mean, $\lim_{n \rightarrow \infty} \int_X n \log \left( 1 + \frac{|f(x)|^2}{n} \right) \,dm \neq 0.$ $\endgroup$ – Karatug Ozan Bircan Jan 6 '13 at 19:43
  • $\begingroup$ Absolutely. (Except if $f=0$ almost everywhere.) $\endgroup$ – Did Jan 6 '13 at 19:49
  • $\begingroup$ If $f$ is not in $L^2$, the the limit is still $\int_X|f|^2=+\infty$. $\endgroup$ – Julien Jan 6 '13 at 20:38
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    $\begingroup$ Try $|f(x)|=n$. $\endgroup$ – Did Jan 6 '13 at 20:48

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