Exercise 7, § 6

Let $f$ be a numerical function that is positive, measurable and non-negligible. a) Show that if, for a finite number $r > 0$, $f^r$ is integrable, then $\log f$ is quasi-integrable (§ 5, Exer. 15). b) Suppose that $f^r$ is integrable for $0 < r < r_0$. Let $A$ be the set of $x \in X$ such that $f(x) > 0$; show that if $\mu^*(A) > 1$, then $N_r(f)$ tends to $+\infty$ as $r$ tends to 0; if $\mu(A) < 1$, $N_r(f)$ tends to 0 with $r$ (make use of Prop. 4 of Ch. I). c) If $\mu(A) = 1$, show that $\int f^r d\mu$ tends to 1 as $r$ tends to 0, and has a right-derivative at this point, equal to $\int \log f \, d\mu$ (make use of Exer. 12 of § 5); from this, deduce that as $r$ tends to 0, $N_r(f)$ tends to $G(f) = \exp(\int \log f \, d\mu)$. d) If $\mu(X) = 1$ and if $f^r$ and $\log f$ are integrable, show that $G(f) \leq N_r(f)$, the equality holding only if $f$ is constant almost everywhere (make use of Exer. 4). e) If $\mu(X) = 1$ and if $f$ and $g$ are two measurable functions, positive and such that $G(f)$ and $G(g)$ are defined, show that $G(f + g)$ is defined and that $G(f) + G(g) \leq G(f + g)$, the equality holding only if there exist two numbers $\alpha, \beta$, not both zero, such that $\alpha f(x) = \beta g(x)$ almost everywhere, or if $G(f + g) = 0$ (make use of $d$), considering the functions $f/(f + g)$ and $g/(f + g)$).