I have found this exercise in the part of Lebesgue integration of my notes of Functional Analysis:
Let we consider $1<p<\infty$ and $q$ such that $\frac{1}{p}+\frac{1}{q}=1$. If $f$ is a measurable function in $\mathbb{R}^n$ verifying $$\frac{\int_A{|f|}}{\left( \int_A1\right)^{1/q}}<\infty\;\forall A\text{with non-zero measure},$$ prove that the set of real numbers $\left \{ \frac{\int_A{|f|}}{\left(\int_A1\right)^{1/q}}:A\text{ has positive measure}\right\}$ has a maximun.
I am very lost with this exercise, I know that the set admits a supremun because it is bounded, but I don't know why it has a maximun. Can someone give me a hint about this?