Shape of $\{a>0:\int |f|^a \text{d}\mu<\infty\}$

Question

Given a measure space $\mathcal{M}=(X,\mathcal{A},\mu)$ and a measurable function $f:X\to\mathbf{R}$, what shapes can the following set take? $$\{a>0:\int |f|^a \text{d}\mu<\infty\}$$ Is it always a half-open intervall, can it contain isolated points, etc.?

Answer 1

Let the set be denoted $S$. If $\mu(X) < \infty$, then the set has the property that $a \in S$ and $b < a$ implies $b \in S$. The example $f(x) = 1/x$ on $(0, 1]$ shows $S$ can be of the form $(0, p)$. Similarly the example $f(x) = 1/(x \log x)$ shows $S$ can be of the form $(0, p]$.

If $\mu(X) = \infty$, let $A := \{f \lt 1\}$ and $B:= \{f \geq 1\}$. Suppose $a < b < c$, where $a, c \in S$, then we clearly have $$\int_A |f|^a d \mu \geq \int_A |f|^b d \mu \geq \int_A |f|^c d \mu,$$ and similar inequalities for $\int_B$. This shows that $b \in S$ as well.

Now insert appropriate combinations of $1/x$ or $1/(x \log x)$ in $f$ shows that $S$ can be of any interval type: $(p, q)$, $[p, q)$, $(p, q]$, and $[p, q]$.