Exercise 6, § 6

For every measurable and non-negligible numerical function $f \geq 0$, show that $N_r(f)$ is an indefinitely differentiable function of $r$ at every point interior to the interval where it is finite. From this, deduce that in the interior of the interval where $N_r(f)$ is finite, $\log N_r(f)$ is a strictly convex function of $1/r$ provided that $f$ is not almost everywhere constant in the set of $x \in X$ where $f(x) \neq 0$.