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I got this question on a measure theory exam today, and after hours of discussing with my colleagues, im still quite confused. I have been able to prove the first point, but I am having trouble with points 3 and 4. Hope anyone here can provide some help :)

Let $F(y) = \int_{[0,1]} (1+|f|)^y dx$, need to prove:

  1. Show that $F$ is well defined and continuous in $[0,1]$.
  2. Show that $F$ is derivable in $(0,1)$, and compute its derivative.
  3. Compute $$ \lim_{y \to 0^+} \int_{[0,1]} (1+|f|)^y \log(1 + |f|) dx$$
  4. Finally, conclude that $$ \lim_{y \to 0^+} \left(\int_{[0,1]} (1+|f|) dx \right)^{\frac{1}{y}} = e^{\int_{[0,1]} \log(1 + |f|) dx}$$
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    $\begingroup$ For point 3, $\frac{\partial}{\partial y} (a^{y})=a^{y} \ln(a)$. Limits should be interchangable, probably using ideas from the bits above, so it will just be the limit $F'(0^{+})$. The same as the integral of just the logarithm term. And for part 4, consider $\lim_{y\to 0^{+}} \frac{\ln(F(y))}{y}$. Use L'Hôpital's rule, then exponentiate. $\endgroup$
    – Darmani V
    Commented Jun 14 at 8:44

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