习题 4, § 6

设 $\varphi(t_1, t_2, \ldots, t_n)$ 为第 1 章命题 1 的有限数值函数满足条件。 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). 特别是，$p$和$q$表示两个共轭指数，使得$1 < p < +\infty$： $1^\circ$对于两个正数函数$f \in \mathscr{L}^p, g \in \mathscr{L}^q$使得$\int fg d\mu = N_p(f)N_q(g)$，存在两个数$\alpha, \beta$是充分必要的，而不是同时存在零，使得$\alpha(f(x))^p = \beta(g(x))^q$几乎无处不在； $2^\circ$对于两个正数函数$f \in \mathscr{L}^p, g \in \mathscr{L}^p$使得$N_p(f+g) = N_p(f) + N_p(g)$，存在两个数$\alpha, \beta$是必要且充分的，而不是同时存在零，使得$\alpha f(x) = \beta g(x)$几乎无处不在。