Let $\mu$ be a (positive real) measure on a space $X$ and $f$ a measurable function. Put $F: (0,\infty) \rightarrow \overline{\mathbb R}_+, p \mapsto \int_X |f|^p ~d\mu$. Write $D = \{ p \in (0,\infty) \mid F(p) < \infty\}$.
It is easy to see that $F(p) \le F(r) + F(s)$ if $r<p<s$, whence the fact that $D$ is connected alias an interval. Moreover, $\log F$ is convex by Hölder's inequality.
- Is there $\mu$ and $f$ such that $D$ has exactly one element?
- The example of $\mu$ Lebesgue-measure on $(0,1)$ resp. $(1,\infty)$ and $f(x) = x^\alpha$ shows that $D$ can be an open interval. Can it be closed? Left-open, right-closed? Left-closed, right-open?
In my intuition, $D$ should be open, at least left-open in general. What do you think?
This is an (as I guess, standard) exercise, taken from the Big Rudin. I apologize if this is a duplicate, thank you for indicating the answering article then.
It's a fun exercise. It is mainly based on two measures $\mu_1,\mu_2$ defined on $\mathbb{R}$ as follows : $$ \mu_1= \sum_{n=2}^{\infty} \frac{1}{n^2ln(n)^2}\delta_{n}$$ and $$ \mu_2= \sum_{n=2}^{\infty} \frac{1}{ln(n)^2} \delta_{\frac{1}{n}}$$ With $f(x)=x$ and $\mu:= \mu_1 +\mu_2$. We see clearly that :
$$ D^{\mu}_f = \{1\}$$ (which is also a closed set) $$ D^{\mu_1}_f = (0,1]$$ $$ D^{\mu_2}_f= [1,\infty)$$