Exercise 4, § 6

Let $\varphi(t_1, t_2, \ldots, t_n)$ be a finite numerical function satisfying the conditions of Prop. 1 of Ch. I. Show that if $f_1, f_2, \ldots, f_n$ are $n$ finite numerical functions that are positive, integrable and non-negligible, then the function $\varphi(f_1, f_2, \ldots, f_n)$ is integrable and

Moreover, in order that it is necessary and sufficient that, for almost every $x \in X$, the point of $\mathbf{R}^n$ whose coordinates are $\xi_i = f_i(x)/\varphi(f_1(x), \ldots, f_n(x))$ belong to the facet with respect to $K$ of the point whose coordinates are (argue as in Exer. 2). In particular, $p$ and $q$ denoting two conjugate exponents such that $1 < p < +\infty$: $1^\circ$ for two positive numerical functions $f \in \mathscr{L}^p, g \in \mathscr{L}^q$ to be such that $\int fg d\mu = N_p(f)N_q(g)$, it is necessary and sufficient that there exist two numbers $\alpha, \beta$, not both zero, such that $\alpha(f(x))^p = \beta(g(x))^q$ almost everywhere; $2^\circ$ for two positive numerical functions $f \in \mathscr{L}^p, g \in \mathscr{L}^p$ to be such that $N_p(f+g) = N_p(f) + N_p(g)$, it is necessary and sufficient that there exist two numbers $\alpha, \beta$, not both zero, such that $\alpha f(x) = \beta g(x)$ almost everywhere.