Let $ f \in L^1[0,1]$ be a real valued Lebesgue integrable function on $[0,1]$. Prove that

$$ \lim_{n \rightarrow \infty } \frac{1}{n} \int_0^1 \ln \big( 1 + e^{nf(t)} \big) dt = \int_0^1 f^+(t) dt \, , $$ where $f^+(t) =\max (0,f) $.

I feel like I should apply the dominated convergence theorem somehow but I can't see a substitution that makes this work. Any suggestions, hints would be greatly appreciated.

  • 1
    $\begingroup$ Consider that for $t$ where $f(t)$ is not zero but is positive, you can essentially ignore the $1+$ in the log. Where $f\leq0$, you can see that you get zero in the numerator. This is not rigorous, but maybe provides some helpful insight. $\endgroup$
    – jdods
    Jul 21 '16 at 15:58
  • $\begingroup$ Very helpful! Thank you! $\endgroup$
    – user209663
    Jul 21 '16 at 17:03

I leave to you the proof that $g_n(t)=\frac{1}{n}\log (1+\exp(nf(t)) \to f^+(t)$ for fixed $t$.

Now $\exp(nf(t))\leq \exp(n|f(t)|)$ and then $$0\leq g_n(t)\leq \frac{1}{n}\log (1+\exp(n|f(t)|)=|f(t)|+\frac{1}{n}\log (1+\exp(-n|f(t)|))$$ and then $$0\leq g_n(t) \leq |f(t)|+\log(2)=h(t)\in L^1$$

Now you can use the DCT.


Write $[0,1]= ([0,1]\cap\{f>0\})\cup ([0,1]\cap\{f\leq 0\})=A\cup B$.

Define $g_n(t)=ln(1+(e^{f(t)})^n)/n$. Over A, $g_n\to f(t)$ and over B $g_n\to 0$.

Apply BeppoLevy and mayorated, respectly.

  • $\begingroup$ I think that over $A$, $g_n(t)\to f(t)$ $\endgroup$
    – Svetoslav
    Jul 21 '16 at 16:32
  • $\begingroup$ @Svetoslav yes! Tks $\endgroup$ Jul 21 '16 at 16:36

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