I have reviewed Ayman Houreih's proof for the limit of the $L_p$ norm as $ p \rightarrow 0$ at "Scaled $L^p$ norm" and geometric mean.
While I have found the outline of the proof very helpful, I have a question regarding the applicability of LDCT:
The proof mentions $\frac{|f|^q - 1}{q}$ as the dominating function. A comment in the proof also mentions that $\frac{|f|^q - 1}{q}$ decreases as $x \rightarrow 0$. I have the following question:
I don't belive $\frac{|f|^q - 1}{q}$ decreases as $x \rightarrow 0$ when $|f| < 1$. This can easily be verified by computing the the value of $\frac{|f|^x - 1}{x} $ for different values of $x$ when $|f| < 1$. Am I missing something obvious? Can someone please share what the dominating function (say $g$) is and explain how $$ |\frac{f_n^{(1/n)} - 1}{(1/n)}| \leq g $$ where $g$ is a real Lebesgue integrable function?
Thanks.